Systems of Linear Equations and Matrices-Linear Algebra-Lecture Slides, Slides of Linear Algebra

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Linear Algebra

Systems of Linear Equations

and Matrices

2

Matrices

 Variety of engineering problems lead to the need to solve systems of linear equations

Ax  b



 

 

 

  m m m mn

nn a a a a

aa aa aa aa   

 1 2 3

A^11211222132321 

 

 

 

  b n

b bb ^21 

 

 

 

  x n

x xx ^21

where

matrix column vectors

4

Examples

 These are valid matrices

 01 10   ca eb  ^101203 

 These have problems

^132   5 4 6  

 

 

 

 8109

(^117)

5

Matrix Notations

 Denote matrices by Capital Boldface Letters A,B,C, … or by writing the general entry in

brackets likeand n columns can also be written as: A [ ajk ]. A matrix A with m rows

(^1121 1222 1323 ) 1 2 3

nn m m m mn

aa aa aa aa

a a a a

   ^ 

jk  

A a

7

Row and Column Matrices

(vectors)

 ROW MATRIX one row, i.e., the dimension (or row vector m ) is a matrix with = 1

r  r 1 r 2 r 3 rn 

 COLUMN VECTOR column is a matrix with only one

1 2 m

c

c

c

 ^ 

c

8

Square Matrix

 When the row and column dimensions of a matrix are equal ( m = n ) then the matrix is

called square



 

 

 

 

n n nn

n

n

a a a

a a a a a a   



A

10

Diagonal Matrices

 Diagonal Matrix

 Tri-diagonal Matrix

a nn

a a

A^011022 ^0000



 

 

 

  54 5545

21 3222 23 11 12 00 00 0

0 0 00

0 0 0 a aa

a aa a a a A  

11

Identity Matrix



 

 

 

  00 00 01 10

 Identity Matrix I 01 10 00 00

IA  AI  A

 The identity matrix has the property thatif A is a square matrix, then

13

Example - Transpose

A  (^)  01 13 04    

 

  43 10 A^ T^10

A  (^) ^1253  A T ^  31 52 

14

Matrix Equality

 Two ( and only if they have the same size and each m x n ) matrices A and B are equal if

of their elements are equal. That is

A if and only if = B

ajk = b j k for j = 1,..., m ; k = 1,..., n

16

Examples - Vector Addition



 

 

 

  42

u^13 

 

 

 

 

 21

v^53



 

 

 

  

 

 

 

 

    

 

 

 

    

 

 

 

   2 12

4 4 2 23 15

1 3 2

(^51) 3 4 23

1 u v

17

Matrix Addition

11 12 1 11 12 1 21 22 1 21 22 1 1 2 1 2 11 11 12 12 1 1 21 21 22 22 1 1 1 1 2 2

n n n n m m mn m m mn n n n n m m m m mn mn

a a a b b b a a a b b b a a a b b b a b a b a b a b a b a b a b a b a b

      ^ ^ ^       ^ ^   ^ ^   ^ ^ 

A B

    

19

Scalar – Matrix Multiplication

 Multiplication defined as of a matrix A by a scalar is



 

 

 

  m m mn

nn a a a

aa aa aa

 

 1 2

A^1121122212

 Examples

  4 , A  01 12 ,  A  04 48 

   2 , A ^120413 ,  A  42  08  62  Docsity.com

20

Matrix – Matrix Multiplication

 The the number of columns of product of two matrices A A is equal to the number of and B is defined only if

 rows ofIf A is (^ m B .x p ) and B is ( p x n ), the product is an ( m x

n ) matrix C

C m  n  A m  p B p  n