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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= 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 811.
§1.2 Linear Combinations, MatrixVector 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
matrixvectorproduct 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 θrotation 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 .
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