# Matrices, Vectors and Systems of Linear Equations

**§1.1 Matrices and Vectors
I Matrices
**

**Definition 1. A matrix**is a rectangular array of scalars. If the matrix has m rows and

n columns, we say that the

**size**of the matrix is m by n, written m × n. The matrix is

**square**if m = n. The scalar in the ith row and jth column is called the (i, j)-

**entry**of

the matrix.

Remark 1.

(i) Two matrices A and B are **equal** if they have the same size and
for all i

and j.

(ii) If A and B are m × n matrices and c is a scalar, then

• A± B is the m × n matrix with the (i, j)-entry=
.

• cA is the m × n matrix with the (i, j)-entry= ca_{ij} .

• The transpose A^{T} of A is the n×m matrix with the (i, j)-entry= the (j, i)-entry

of A, that is,
.

(iii) Special matrices:

• ** Zero matrix ** 0 is the matrix with each entry= 0.

• The n × n **identity matrix** is the matrix with the diagonal entry= 1 and the

rest of entries are zero.

**Theorem 1. Properties of Matrix Addition and Scalar Multiplication
**

Let A, B and C be m × n matrices, and let s and t be any scalars. Then

(a) A + B = B + A. (commutative law of matrix addition)

(b) (A + B) + C = A + (B + C). (associative law of matrix addition )

(c) A + 0 = A.

(d) A + (-A) = 0.

(e) (st)A = s(tA).

(f) s(A + B) = sA + sB.

(g) (s + t)A) = sA + tA.

**Theorem 2. Properties of the Transpose**

Let A and B be m × n matrices, and let s be any scalars. Then

**II Vectors**

A matrix that has exactly one row is called a

**row vector**and a matrix that has exactly

one column is called a

**column vector**. The entries of a vector are called

**components**.

Remark 2. Geometrical interpretation of vectors are explain in the textbook, page 8-11.

§1.2 Linear Combinations, Matrix-Vector Products and Special

Matrices

III Linear Combinations

§1.2 Linear Combinations, Matrix-Vector Products and Special

Matrices

III Linear Combinations

**Definition 2. A linear combination**of vectors is a vector of the form

where are scalars. These scalars are called the

**coefficients**of the linear

combination .

Remark 3.

(i) In general, the

**standard vector**of R

^{n}is defined by

(ii) If and
are any nonparallel vector in R^{2}, then
every vector in R^{2} is a linear

combination of and
.

**IV Matrix- Vector Products **

**Definition 3.** Let A be an m × n matrix and be an n × 1 vector. We define the

**matrix-vector-product** of A and , denoted by A , to be the linear
combination of

the column of A whose coefficients are the corresponding components of
. That
is,

**Theorem 3. Properties of Matrix- Vector Products **

Let A, B and C be m × n matrices, and let
and
be vectors in R^{n}. Then

for every scalar c.

where is the jth standard vector
in R^{n}.

(e) If B is an m × n matrix such that for all
in R^{n}, then B = A.

(f) is m × 1 zero vector .

(g) If 0 is the m × n zero matrix, then 0 is the m × 1 zero vector.

Remark 4. If follows from (a) and (b) in the above theorem, that for any m × n
matrix

A, any scalars and any vector
in R^{n},

**V Rotation Matrices in R ^{2}**

**Definition 4.**In R

^{2}, we call

the θ-rotation matrix, or more simply , a rotation matrix.

Remark 5.

1. For any vector in R^{2}, the vector is the vector obtained by rotating
by

an angle θ, where the rotation is counterclockwise if θ > 0 and clockwise if
θ <
0.

2. The 0^{0}-rotation matrix .

Prev | Next |