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Abstract Linear Algebra
Linear Algebra. Session 8
Dr. Marco A Roque Sol
08/01/2017
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let
V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W
be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and
L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W,
be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range
(or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image)
of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L
is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set
of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors
w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W
suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat
w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v)
for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some
v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V.
The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L
is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel
of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L,
denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L)
is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set
of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors
v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ V
such that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that
L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Range and kernel
Let V,W be vector spaces and L : V→W, be a linear mapping.
Definition.
The range (or image) of L is the set of all vectors w ∈W suchthat w = L(v) for some v ∈ V. The range of L is denoted by L(V)
The kernel of L, denoted by ker(L) is the set of all vectors v ∈ Vsuch that L(v) = 0.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider
the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation,
L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3,
given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
=
1 0 −11 2 −11 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AX
Find ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind
ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and
range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of
L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel,
ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L),
is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace
of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A.
To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To find
ker(L), we apply row reduction to the matrix A: 1 0 −11 2 −11 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L),
we apply row reduction to the matrix A: 1 0 −11 2 −11 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction
to the matrix A: 1 0 −11 2 −11 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A:
1 0 −11 2 −11 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒
1 0 −10 2 00 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒
1 0 −10 1 00 0 0
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.1
Consider the linear tranformation, L : R3 → R3, given by
L
xyz
= 1 0 −11 2 −1
1 0 −1
= xy
z
= AXFind ker(L) and range of L
Solution
The kernel, ker(L), is the nullspace of the matrix A. To findker(L), we apply row reduction to the matrix A: 1 0 −11 2 −1
1 0 −1
⇒ 1 0 −10 2 0
0 0 0
⇒ 1 0 −10 1 0
0 0 0
⇒Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence
(x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z)
belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L)
if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0.
It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows that
ker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L)
is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line
spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since
L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L
is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
=
x
111
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+
y
020
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+
z
−1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then,
The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range
of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L,
is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned
by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1),
(0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and
(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1).
It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that
L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3)
is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane
spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Hence (x , y , z) belongs to ker(L) if x − z = y = 0. It follows thatker(L) is the line spanned by (1, 0, 1)
Since L is given by
L
xyz
= x 11
1
+ y 02
0
+ z −1−1−1
then, The range of L, is spanned by vectors (1, 1, 1), (0, 2, 0), and(−1,−1,−1). It follows that L(R3) is the plane spanned by(1, 1, 1), (0, 2, 0).
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider
the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation,
L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R),
given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′.
Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find
range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L)
of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According
to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory
of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations,
the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x);
u(x0) = u0; u′(x0) = u
′0; u′′(x0) = u
′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution
for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and
any u0, u′0, u′′0 . It
Follows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 .
ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that
L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Example 8.2
Consider the linear tranformation, L : C (R)3 → C (R), given byL(u) = u′′ − 2u′ + u′. Find range and ker(L) of L
Solution
According to the theory of differential equations, the initial valueproblem
u(x)′′−2u(x)′+u(x)′ = g(x); u(x0) = u0; u′(x0) = u′0; u′′(x0) = u′′0
has a unique solution for any g ∈ C (R) and any u0, u′0, u′′0 . ItFollows that L(C (R)3) = C (R)
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also,
the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation
l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 )
which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is
alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping
l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3,
becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible
whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to
ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L).
Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence
dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now,
the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1
satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and
W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0.
Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,
ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation
is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation
of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where
L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W,
is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping,
b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b
is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector
from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and
x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x
is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector
from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
Also, the initial data evaluation l(u) = u → (u0, u′0, u′′0 ) which is alinear mapping l(u) : C (R3)→ R3, becomes invertible whenrestricted to ker(L). Hence dim (ker(L)) = 3
Now, the functions xex , ex , 1 satisfy L(xex) = L(ex) = L(1) = 0and W (xex , ex , 1) 6= 0. Therefore,ker(L) = Span(xex , ex , 1)
General linear equations
Definition.A linear equation is an equation of the form
L(x) = b
where L : V→W, is a linear mapping, b is a given vector from W,and x is an unknown vector from V.
Dr. Marco A Roque Sol Linear Algebra. Session 8
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
The range of L is the set of all vectors b ∈W such that theequation L(x) = b has a solution.
The kernel of L is the solution set of the homogeneous linearequation L(x) = 0 .
Theorem
If the linear equation L(x) = b is solvable and dim (ker(L))
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
The range
of L is the set of all vectors b ∈W such that theequation L(x) = b has a solution.
The kernel of L is the solution set of the homogeneous linearequation L(x) = 0 .
Theorem
If the linear equation L(x) = b is solvable and dim (ker(L))
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
The range of L
is the set of all vectors b ∈W such that theequation L(x) = b has a solution.
The kernel of L is the solution set of the homogeneous linearequation L(x) = 0 .
Theorem
If the linear equation L(x) = b is solvable and dim (ker(L))
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
The range of L is the set
of all vectors b ∈W such that theequation L(x) = b has a solution.
The kernel of L is the solution set of the homogeneous linearequation L(x) = 0 .
Theorem
If the linear equation L(x) = b is solvable and dim (ker(L))
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
The range of L is the set of all vectors
b ∈W such that theequation L(x) = b has a solution.
The kernel of L is the solution set of the homogeneous linearequation L(x) = 0 .
Theorem
If the linear equation L(x) = b is solvable and dim (ker(L))
Abstract Linear AlgebraRange and kernel.Matrix transformations. Matrix of a linear transformation. Similar matrices.
Range and kernel.
The range of L is the set of all vectors b ∈W
such that theequation L(x) = b has a solution.
The kernel of L is the solution set of the homogeneous linearequation L(x) = 0 .
Theorem
If