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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