MATH 17100
28446
11/02/2009
Matrices
Recall the
vectors
we defined as
n
tuples, namely , ordered collections of numbers, for which we
devised and defined operations of addition, multiplication by a scalar, dotmultiplication and
crossmultiplication, using them in
spatial geometry
to describe lines and curves, planes and
surfaces. We now continue to extend and expand the use of vectors in solving systems of linear
equations in
algebra
.
We first adopt some new forms of expression for the vector.
Given the vector with
n
components,
12
(
,
,
,
)
n
v v
v
v
, we may express the same quantity as
[
]
n
v v
v
and call it a
row vector
.
We may also express it as
1
2
n
v
v
v
and call it a
column vector
. Notice that the commas delineating
the components in our previous notation have been dropped, the components placed within
square parentheses are separated by blank spaces when the components are listed
horizontally
in
a
row
, and the vector is called a row vector, or they are placed on separate lines along a
vertical
column
, and the vector is called a column vector. With this distinction, equality and addition of
vectors require that they be
compatible
, namely that they have the same number of components
and being of the same type, i.e. row or column.
Consider the row vector with
n
components. Suppose each scalar component were replaced by a
column vector with
m
components:






n
v
v
v
,
,
1, 2,
, ,
k
kn
v
being the
th
k
column vector along the row,
11
1
21
22
2
n
n
m
m
m n
v
v
v
v
v
v
v
v
v
where, expressing explicitly the components of each column vector
k
v
, we find that
ik
v
is the
th
i
component of the
k
column vector, or, judging from the
rectangular array of quantities, we can say that
v
is the quantity found at the intersection of the