CHAPTER 4LINEAR TRANSFORMATION AND MATRICES

Definition 4.1Let V and W be vector spaces. A linear transformation L of V into W is a function assigning a unique vector L(X) in W to each X in V such that:

(a) ,for every X and Y in V.(b) , for every X in V and every scalar c.

Example: 1.The function defined by L(x,y) =(2x,y) is a linear transformation.

2. The function defined by L(x,y) =(x,y+1) is not a linear transformation.

3. Let function be the transformation that maps f into its derivative; that is D(f) = f is a linear transformation.

Theorem 4.1Let is a linear transformation, then

for any vectors in V and scalars.

Theorem 4.2 Let is a linear transformation. Then a)L(Ov) = Ow, where Ov and Ow are the zero vectors in V and W, respectively.L(X Y) = L(X) L(Y)Proof:a.Ov = Ov +OvL(Ov)= L(Ov +Ov)= L(Ov) +L(Ov) b.L(x - y) = L(x + -y) = L(x) + L(-y) = L(x) L(y)

Theorem 4.3 Let be a linear transformation of an n-dimensional vector space V into a vector space W. Also let S= {X1, X2, , Xn}be a basis for V. If X is any vector in V, then L(X) is completely determined by {L(X1), L(X2), ,L(Xn)}.

Proof: Let X be in V then where c1, c2, , cn are uniquely determined real numbers.

by Theorem 4.1.Thus, L(X) is completely determined by the elements L(x1), L(x2),, L(xn).

Example. Let L: R3R2 be a linear transformation for which we know that L(1,0,0) = (2,-4), L(0,1,0) = (3, -5), L(0,0,1) = (2,3).

What is L(-1,-2,3)?b.What is L(a,b,c)?

Note that {(1,0,0), (0,1,0), (0,0,1)} is a basis for R3 so that (-1,-2,-3)= -1(1,0,0) 2(0,1,0) + 3(0,0,1)By Theorem 4.3:a. L(-1,-2,-3) = -1 L(1,0,0) -2L(0,1,0) +3L(0,0,1) = -1(2,-4) -2(3,-5) + 3(2,3) = -(2,-4) +(-6,10) + (6,9) = (-2,23)

b. (a, b, c) = a(1,0,0) + b(0,1,0) + c(0,0,1) L(a, b, c) = a(2,4) + b(3,-5) + c(2,3)= (2a + 3b + 2c, -4a 5b + 3c)

2. L: R2R2 L = L =

a.What is L?

Solution:

= 3 - 5

L = 3 - 5

=

EDMODOReminder: Code: sbwczkCourse: Math120-Y

Definition 4.2: A linear transformation L: VW is said to be one-one if for all X1, X2, in V, X1 X2 implies that L(X1)L(X2). An equivalent statement is that L is one-to one if for all X1, X2 in V, L(X1)=L(X2) implies that X1=X2. If L(x1) = L(x2) x1 = x2

Example: Let L: R2R3 be defined by L(x,y) = (x, x+y, y) Is L one one? Let x1 = (x1,y1) x2 = (x2,y2) Suppose L(x1) = L(x2) (x1 , x1 + y1, y1) = (x2, x2 + y2, y2)x1= x2y1= y2 L is one one.

2. L: R3R2 defined by L(x,y,z) = (x,0)L(3,4,-2) = (3,0)L(3,4,1) = (3, 0)(3,4,-2) (3,4,1)L(3,4,-2) = L(3,4,1) Therefore, it is not one to one function.

Definition 4.3: Let: L: VW is be linear transformation. The kernel of L, ker L, is the subset of V consisting of all vectors X such that L(X)=Ow.

Example: Let L: R3 R3 be a linear transformation defined by

Determine the Ker L.

If L:R4 R2 is defined by

is a linear transformation , find the kernel L.

Theorem 4.4If L: VW is a linear transformation, then kernel L is a subspace of V.

Theorem 4.5A linear transformation L: VW is one-to-one if and only if ker L={0v}

Proof: Suppose L is one-one. Let X Ker L.

Suppose Ker L={ov}. Assume that L(X1)= L(X2), for X1, X2 V.

Definition 4.4 If L: VW is a linear transformation, then the range of L, denoted by range L, is the set of all vectors in W that are images, under L, of vectors in V. Thus a vector Y is in range L if we can find some vector X in V such that L(X)=Y. If range L=W, we say that L is onto.

Example1. Let L: R2 R3 be defined by L(x,y) = (x, x+y, y) Is L onto?

Solution:Given any , find

such that L(X)= Y.

Example 2: Let L: R4 R3 be a linear transformation defined by . Is L onto?

**************