m211f2012 Slides Sec1.3 2.1handout

7/28/2019 m211f2012 Slides Sec1.3 2.1handout

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MATH 211 Winter 2013

Lecture Notes(Adapted by permission of K. Seyffarth)

Sections 1.3 & 2.1

Sections 1.3 & 2.1 Page 1/1


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1.3 Homogeneous Equations



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Homogeneous Equations

Example

Solve the systemx1 + x2 x3 + 3x4 = 0x1 + 4x2 + 5x3 2x4 = 0

x1 + 6x2 + 3x3 + 4x4 = 0

1 1 1 3 0

1 4 5 2 01 6 3 4 0

1 0 95

145 0

0 14

5

1

5 00 0 0 0 0

The system has infinitely many solutions, and the general solution inparametric form is

x1 = 95 s 145 tx2 =

45 s

15 t

x3 = sx4 = t

or

x1x2x3x4

=

95 s 145 t45 s

15 t

s

t

, where s, t R.



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Definition

If X1, X2, . . . , Xp are columns with the same number of entries, and ifk1, k2, . . . kr R (are scalars) then k1X1 + k2X2 + + kpXp is a linear

combination of columns X1, X2, . . . , Xp.

Example (continued)

In the previous example,

x1x2x3x4

=

95 s

145 t

45 s15 t

s

t

=

95 s

45 s

s

0

+

145 t

15 t

0t

= s

9545

10

+ t

14515

01



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Example (continued)

This gives us

x1x2x3x4

= s

9

545

10

+ t

14

515

01

= sX1 + tX2,

where X1 =

9545

10

and X2 =

145

1501

.

The columns X1 and X2 are called basic solutions to the originalhomogeneous system.

The general solution to a homogeneous system can be expressed as alinear combination of basic solutions.



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Example (continued)

Notice that

x1x2x3x4

= s

9545

10

+ t

14515

01

=s

5

9450

+t

5

14105

= r

9450

+ q

14105

= r(5X1) + q(5X2)

where r, q R.



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Example (continued)

The columns 5X1 =

945

0

and 5X2 =

1410

5

are also basic solutions

to the original homogeneous system.

In general, any nonzero multiple of a basic solution (to a homogeneoussystem of linear equations) is also a basic solution.



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1.3, Exercise 2(c)

Find all values ofa

for which the systemx + y z = 0

ay z = 0x + y + az = 0

has nontrivial solutions, and determine the solutions.

When a = 0,

xy

z

= s

11

0

, s R, and when a = 1,

xyz

= s 211

, t R.



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2.1 Matrix Addition, Scalar Multiplication andTransposition



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Matrices - Basic Definitions and Notation

Definitions

Let m and n be positive integers.

An m n matrix is a rectangular array of numbers having m rows andn columns. Such a matrix is said to have size m n.

A row matrix is a 1 n matrix, and a column matrix (or column) isan m 1 matrix.

A square matrix is an m m matrix.The (i,j)-entry of a matrix is the entry in row i and column j.

General notation for an m n matrix, A:

A =

a11

a12

a13

. . . a1na21 a22 a23 . . . a2n

a31 a32 a33 . . . a3n...

......

...am1 am2 am3 . . . amn

= [aij]



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Matrices - Properties and Operations

1 Equality: two matrices are equal if and only if they have the samesize and the corresponding entries are equal.

2 Addition: matrices must have the same size; add correspondingentries.

3 Scalar Multiplication: multiply each entry of the matrix by thescalar.

4 Zero Matrix: an m n matrix with all entries equal to zero.

5 Negative of a Matrix: for an m n matrix A, its negative is

denoted A and A = (1)A.6 Subtraction: for m n matrices A and B, A B = A + (1)B.



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Matrix form for solutions to linear systems

Example

The reduced row-echlon form of the augmented matrix for the system

x1 2x2 x3 + 3x4 = 12x1 4x2 + x3 = 5x1 2x2 + 2x3 3x4 = 4

is 1 2 0 1 20 0 1 2 1

0 0 0 0 0

leading to the solution

x1 = 2 + 2s tx2 = sx3 = 1 + 2tx4 = t

s, t R.



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Matrix form for solutions to linear systems

Example (continued)

x1 = 2 + 2s tx2 = sx3 = 1 + 2tx4 = t

can be expressed as

x1x2x3x4

=

2 + 2s ts

1 + 2tt

. But

2 + 2s ts

1 + 2tt

=

2010

+ s

2100

+ t1021

Therefore,

x1x2x3x

4

=

2010

+ s

2100

+ t

1021

, s, t R.



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Matrix Addition and Scalar Multiplication

Theorem (2.1 Theorem 1)

Let A, B and C be m n matrices, and let k, p R.

1 A + B = B + A

2 (A + B) + C = A + (B + C)

3

There is an m n matrix 0 such that A + 0 = A and 0 + A = A.4 For each A there is an m n matrixA such that A + (A) = 0 and

(A) + A = 0.

5 k(A + B) = kA + kB

6

(k + p)A = kA + pA7 (kp)A = k(pA)

8 1A = A



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Matrix Transposition

DefinitionIf A is an m n matrix, then its transpose, denoted AT, is the n m

matrix whose ith row is the ith column of A, 1 i n.

Theorem (2.1 Theorem 2)Let A and B be m n matrices, and let k R.

1 AT is an n m matrix.

2 (AT)T = A.

3 (kA)T

= kAT

.4 (A + B)T = AT + BT.



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Symmetric Matrices

Definition

Let A = [aij] be an m n matrix. The entries a11

, a22

, a33

, . . . are calledthe main diagonal of A.

Definition

The matrix A is called symmetric if and only if AT = A. Note that this

immediately implies that A is a square matrix.

Examples

2 3

3 17

,

1 0 50 2 115 11 3

,

0 2 5 1

2 1 3 05 3 2 7

1 0 7 4

are symmetric matrices, and each is symmetric about its main diagonal.



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Example

Show that if A and B are symmetric matrices, then AT + 2B is symmetric.

Proof.

(AT + 2B)T = (AT)T + (2B)T

= A + 2BT

= AT + 2B, since AT = A and BT = B.

Since (AT

+ 2B)T

= AT

+ 2B, AT

+ 2B is symmetric.


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m211f2012 Slides Sec1.3 2.1handout