Vectors and Matrices
Data is frequently arranged in arrays, that is, sets whose elements
are indexed by one or more subscripts.
If the data consists of numbers, then a one-dimensional array is called a
vector and a two-dimensional array is called a matrix (where the
dimension denotes the number of subscripts). This appendix investigates
these vectors and matrices, and certain algebraic operations involving
them. In this context, the numbers themselves are called scalars.
VECTORS
By a vector u, we mean a list of numbers, say, a1, a2, . . . , an. Such a
vector is denoted by u = (a1, a2, . . . , an)
The numbers ai are called the components or entries of u. If all the ai
= 0, then u is called the zero vector. Two such vectors, u and v, are equal,
written u = v, if they have the same number of components and
corresponding components are equal.
EXAMPLE A.1
(a) The following are vectors where the first two have two components
and the last two have three components:
(3,−4), (6, 8), (0, 0, 0), (2, 3, 4)
The third vector is the zero vector with three components.
(b) Although the vectors (1, 2, 3) and (2, 3, 1) contain the same numbers,
they are not equal since corresponding components are not equal.
Vector Operations
Consider two arbitrary vectors u and v with the same number of
components, say u = (a1, a2, . . . , an) and v = (b1, b2, . . . , bn)
The sum of u and v, written u + v, is the vector obtained by adding
corresponding components from u and v;
that is,
u + v = (a1 + b1, a2 + b2, . . . , an+ bn)
The scalar product or, simply, product, of a scalar k and the vector u,
written ku, is the vector obtained by multiplying each component of u by
k; that is, ku = (ka1, ka2, . . . , kan)
We also define −u = −1(u) and u − v = u + (−v)
and we let 0 denote the zero vector. The vector −u is called the negative
of the vector u.
The dot product or inner product of the above vectors u and v is denoted
and defined by u ・ v = a1b1 + a2b2 +・ ・ ・+anbn
The norm or length of the vector u is denoted and defined by
_u_ =√u ・ u
We note that _u_ = 0 if and only if u = 0; otherwise _u_ > 0.
EXAMPLE A.2 Let u = (2, 3, −4) and v = (1, −5, 8). Then
u + v = (2 + 1, 3 − 5, −4 + 8) = (3,−2, 4)
5u = (5 ・ 2, 5 ・ 3, 5 ・ (−4)) = (10, 15,−20)
−v = −1 ・ (1,−5, 8) = (−1, 5,−8)
2u − 3v = (4, 6,−8) + (−3, 15,−24) = (1, 21,−32)
u ・ v = 2 ・ 1 + 3 ・ (−5) + (−4) ・ 8 = 2 − 15 − 32 = −45
Vectors under the operations of vector addition and scalar multiplication
have various properties, e.g.,
k(u + v) = ku + kv
where k is a scalar and u and v are vectors. Many such properties appear
in Theorem A.1, which also holds for vectors since vectors may be
viewed as a special case of matrices.
Column Vectors
Sometimes a list of numbers is written vertically rather than horizontally,
and the list is called a column vector. In this context, the above
horizontally written vectors are called row vectors. The above operations
for row vectors are defined analogously for column vectors.
A.3 MATRICES
A matrix A is a rectangular array of numbers usually presented in the
form
A =a11 a12 ・ ・ ・ a1n
a21 a22 ・ ・ ・ a2n
...............................
am1 am2 ・ ・ ・ amn
The m horizontal lists of numbers are called the rows of A, and the n
vertical lists of numbers its columns. Thus the element aij , called the ij
entry, appears in row i and column j.We frequently denote such a matrix
by simply writing A = [aij].
A matrix with m rows and n columns is called an m by n matrix, written
m × n. The pair of numbers m and n is called the size of the matrix. Two
matrices A and B are equal, written A = B, if they have the same size and
if corresponding elements are equal. Thus, the equality of two m × n
matrices is equivalent to a system of mn equalities, one for each
corresponding pair of elements.
Amatrix with only one row is called a row matrix or row vector, and a
matrix with only one column is called a column matrix or column vector.
A matrix whose entries are all zero is called a zero matrix and will
usually be denoted by 0.
EXAMPLE A.3
(a) The rectangular array A =1 −4 5
0 3 −2
is a 2×3 matrix. Its rows are [1,−4, 5] and [0, 3,−2], and its
columns are & 1 0 & −4 3 & 5 −2