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Note also that matrix multiplication is noncommutative. That is, we cannot
say in general that AB = BA .
1.4.3 Vector Representation and Transformation
We can represent a vector as a matrix with one column, e.g.,
x 1
x 2
.
x n
x =
,
or with one row, e.g.,
b T = b 1
b m .
b 2
···
In this topic, we will be using column matrices to represent vectors. Should
we want to represent a row matrix, we shall use the transpose, as above. Using
this notation, we can also represent a matrix as its component columns:
A = a 1 a 2
a n .
···
is a mapping that preserves the linear properties of
scale and addition; that is, for two vectors x and y ,
A linear transformation
T
aT ( x )+ T ( y )= T ( a x + y ) .
We can use matrices to represent linear transformations. Multiplying a vector
x by an appropriately sized matrix A , and expanding the terms, we get
b 1
b 2
.
b m
a 11
a 12
···
a 1 n
x 1
x 2
.
x n
a 21
a 22
···
a 2 n
=
.
.
.
.
. . .
a m 1
a m 2
···
a mn
This represents a linear transformation
from an n -dimensional space to an m -
dimensional space. If we assume that both spaces use the standard Euclidean
bases e 1 , e 2 ,..., e n and e 1 , e 2 ,..., e m , respectively, then the column vectors in
matrix A are the transformed basis vectors
T
( e n ).
Multiplying transformation matrices together creates a single matrix that rep-
resents the composition of the matrices' respective transformations. In this way,
we can represent a composition of linear transformations in a single matrix.
T
( e 1 ) ,
T
( e 2 ) ,...,
T
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