A

Linear Algebra Review

Deﬁnition A.1 A matrix A is a rectangular array of numbers. The size of

A is denoted by m × n where m is the number of rows and n is the number

of columns of A. If m = n, then A is called a square matrix.

Example A.2

The following matrices

A =

1 4 −3

−12 9 2

, B =

2

−4

5

, C =

3 1 2

5 2 8

1 7 4

, D =

1 0 −5

have dimension 2 × 3, 3 ×1, 3 × 3, and 1 × 3, respectively. The

matrix C is a square matrix with m = n = 2.

Let A be a m ×n matrix. We denote a

ij

as the element of A that is in the

ith row and jth column. Then, the matrix A can be speciﬁed as A = [a

ij

] for

1 ≤ i ≤ m and 1 ≤ j ≤ n.

Deﬁnition A.3

A vector v of dimension k is a matrix of size k × 1 (a column vector) or

size 1 × k (row vector).

Example A.4

q =

1 −9 3

is a row vector of dimension 1 × 3

w =

9

2

is a column vector of dimension 2 × 1.

Deﬁnition A.5

Given a matrix A, one can generate another matrix by taking the ith row

of A and making it the ith column of a new matrix and so on. The resulting

matrix is called the transpose of A and is denoted by A

T

.

Example A.6

If A =

1 4 −3

−12 9 2

, then A

T

=

1 −12

4 9

−3 2

.

Deﬁnition A.7

A matrix A with the property that A = A

T

is called a symmetric matrix.

Example A.8

323

© 2014 by Taylor & Francis Group, LLC

324 Introduction to Linear Optimization and Extensions with MATLAB

R

If A =

1 −1 −7

−1 2 5

−7 5 3

, then A

T

= A so A is symmetric.

Deﬁnition A.9

A set of vectors V = { v

1

, v

2

, ..., v

l

} each with the same dimension are said

to be linearly independent if

α

1

v

1

+ α

2

v

2

+ ···α

l

v

l

= 0 implies that α

1

= α

2

= ··· = α

l

= 0,

i.e., all scalars are 0. Otherwise the set of vectors V are said to be linearly

dependent.

Deﬁnition A.10

A square m × m matrix A is said to be invertible if there exists a square

m × m matrix B such that AB = I = BA where I is the m × m identity

matrix. B is called the inverse of A and is denoted as B = A

−1

. A matrix A

that has an inverse is said to be invertible of non-singular.

Theorem A.11

A square m×m matrix A is invertible if and only if the m columns (rows)

of A form a linearly independent set of vectors.

The inverse of a square matrix A plays an important role in the solution

of linear systems of equations of the form

Ax = b

since the solution can be represented mathematically as x = A

−1

b.

For instance, the linear system

3x

1

+ 5x

2

= 11

5x

1

− 2x

2

= 8

can be represented in matrix form by letting

A =

3 5

5 −2

, b =

11

8

, and x =

x

1

x

2

,

then the solution is

x = A

−1

b =

3 5

5 −2

−1

11

8

=

2

1

.

The inverse of A is A

−1

=

0.0645 0.1613

0.1613 −0.0968

.

The inverse of square matrices will be obtained in this book via MATLAB

through the solving of the corresponding system of equations. The details for

© 2014 by Taylor & Francis Group, LLC

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