Download Vectors and Vector Operations

1. Vectors and Linear Functions
We begin with vectors and linear functions which provide the framework for the problem
solving methods of linear algebra.
1.1 Vectors
There are various types of objects to which the term vector is applied. These include row
and column vectors, matrices, functions and directed line segments. We begin by looking
at each of these and discuss the features they have in common which is why they are all
called vectors.
Row and column vectors. In many books the term vector is reserved for a list of
 4
numbers. For example, v = -1 is a vector of this type with three components or
 7
elements. The first component is v1 = 4, the second is v2 = -1 and the third is v3 = 7.
 4
When the components of a vector are written in a column as with -1, it is called a
 7
column vector. When they are written in a row as with (4, -1, 7), it is called a row vector.
In some situations we shall distinguish between row and column vectors. The transpose
operation switches between the row and column form of a vector. If v is a certain vector
then
vT
=
transpose of v
=
the same list of numbers as v, but viewed as a row vector if v is
a column vector or viewed as a column vector if v is a row
vector.
For example,
 4 T
-1 = (4, -1, 7)
 7
 4
and (4, -1, 7)T = -1
 7
A row vector (x, y) or column vector (yx) with two components is often called an ordered
pair. A row vector (x, y, z) or column vector y with three components is often called a
z
x
 x12
triple. A row vector (x1, x2, …, xn) or column vector  .  with n components is often
 .
 xn 
called an n tuple.
x
1.1 - 1
5 7
Matrices. A matrix is a rectangular array of numbers. For example, A =  2 3  is
3 5
referred to as a 32 matrix since it has three rows and two columns. The individual
elements of a matrix are referred to by pairs of subscripts, i.e. Aij = the element in row i
and column j of A. In the example above A21 = 2. We shall refer to the rows and
columns of a matrix as follows.
Ai,● = row i of A
A●,j = column j of A
In the example above A2,● = (2, 3) and A●,2 =  3 . We can regard row and column vectors
2
4
as matrices. For example,  1  is a 31 matrix and (-2, 3, 0, 6) is a 14 matrix.
 2
7
The transpose of a matrix A is denoted by AT and obtained by making the rows of A the
columns of the transpose and the columns of A the rows of the transpose. For example,
5 7
A =  2 3 
3 5

5 2 3
AT =  7 3 5 
Thus the element in row i and column j of the transpose is the element in row j and
column i of A, i.e. (AT)ij = Aji. If A is a mn matrix then AT is a nm matrix.
Functions. As we shall see, real-valued functions, such as f(x) = 2x2 – 3e2x and
x
g(x, y) = 4xy - y can also be regarded as vectors. The functions that we deal with most
frequently are defined by means of formulas such as f(x) = 2x2 – 3e2x. However, we
include as functions f any rule that assigns real numbers y = f(x) to every element x of
some set S called the domain of the function. The domain S is part of the definition of the
function. For example, the f(x) = 2x2 – 3e2x could be regarded as a function with domain
S consisting of all real numbers. However, in other application the domain could be
x
taken as some subset of the real numbers. Similarly, g(x, y) = 4xy - y could be regarded
as a function with domain S consisting of all order pairs (x, y) such that y  0. However,
the domain could be taken as any subset of this set.
Row and column vectors and matrices can also be regarded as functions. For example,
 4
v = -1 can be viewed as the function which assigns 4 to the number 1, -1 to the number
 7
2 and 7 to the number 3. Thus the domain of v is S = {1, 2, 3}, the set of integers
1.1 - 2
5 7
consisting of 1, 2, and 3 and v(1) = 4, v(2) = -1 and v(3) = 4. Similarly A =  2 3  can be
3 5
viewed as the function with domain S = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (3, 2)}
consisting of the set of ordered pairs (i, j) where i can be 1, 2 or 3 and j can be 1 or 2.
Then
A(1, 1) = 5
A(1, 2) = 7
A(2, 1) = 2
A(2, 2) = 3
A(3, 1) = 3
A(3, 2) = 5
Row or column vectors whose components are functions or combinations of numbers and
t2
2

sin
functions such as v = t t  or w =  - 1t  are also regarded as vectors. A row or column
 e 
e 
vector whose components are functions with the same domain S can also be regarded as a
function with domain S whose values are row or column vectors. For example, the triple
t2
of functions v =  sint t  can also be regarded as a function f which assigns to each real
 e 
t2
number t the triple of numbers f(t) =  sint t .
 e 
Directed line segments and other things with magnitude and direction. A physicist
will tell you a vector is something with magnitude and direction. A directed line segment

Q
is a good example. If P and Q are two points then PQ represents
the directed line segment from P to Q.
P
Often two directed line segments are regarded as representing the
same vector if they have the same length and direction. If this


Q
is the case we will write PQ = RS if the directed line


segments PQ and RS have the same length and direction.
S
P
Directed line segments are connected to
ordered pairs and triples when one draws
a coordinate system. If the directed line
segments all in lie the same plane, then
we draw a xy-coordinate system for that
x
plane. If the coordinates of P are  y11  we
x
x
shall write P =  y11 . Suppose P =  y11 
x
and Q =  y22 . Then to the directed line
R
y
PQ =
(xy -- xy ) = (xy)
2
1
2
1
Q=
y2
(xy )
2
2
y = y2 – y1
y1
P=
x1
y1
()
x = x2 – x1

segment vector PQ corresponds the
x
x1
1.1 - 3
x2

x -x
x -x
numeric vector  y22 - y11 . We shall indicate this by writing PQ =  y22 - y11 . Note that x =
x2 - x1 is the change in the x coordinate as we move from P to Q and y = y2 - y1 is the

x
change in the y coordinate as we move from P to Q, so that PQ =  y .
Other examples of physical quantities with magnitude and direction are velocities,
accelerations, forces, electric fields and magnetic fields.
The reason row and column vectors, matrices, functions and directed line segments are
all called vectors is because all of these can be added, negated and multiplied by numbers.
These operations are the subject of the next section.
1.1 - 4