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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

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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