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= caij .
• The transpose AT 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

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 Rn is defined by

(ii) If and are any nonparallel vector in R2, then every vector in R2 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 Rn. Then

for every scalar c.

where is the jth standard vector in Rn.

(e) If B is an m × n matrix such that  for all in Rn, 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 Rn,

V Rotation Matrices in R2

Definition 4. In R2, we call

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

Remark 5.

1. For any vector in R2, 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 00-rotation matrix .

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