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Unit 9: Representation of Transformations by Matrices
use of the action of a linear transformation on this basis. Once this much is achieved by means of Notes
the operations in A(V), we can induce operations for the symbols created, making of them an
algebra. This new object, infused with an algebraic life of its own, can be studied as a mathematical
entity having an interest by itself. This study is what comprises the subject of matrix theory.
However to ignore the source of these matrices, that is, to investigate the set of symbol
independently of what they represent, can be costly. Instead we shall always use the interplay
between the abstract, A(V), and the concrete, the matrix algebra, to obtain information one
about the other.
Let V be an n-dimensional vector space over the field F and let W be an m-dimensional vector
space over F. Let 1 ,..., n be an ordered basis for V and ' 1 ,..., m an ordered basis for
W. If T is any linear transformation from V into W, then T is determined by its action on the
vectors . Each of the n vectors T is uniquely expressible as a linear combination
j j
m
T j A ij i ...(1)
i 1
of the i the scalars A ij ,...A being the coordinates of T j in the ordered basis '. Accordingly,
mj
the transformation T is determined by the mn scalars A via the formula (1). The m n matrix A
ij
,
defined by A i j A ij is called the matrix of T relative to the pair of ordered basis and '. Our
immediate task is to understand explicitly how the matrix A determines the linear
transformation T.
If x 1 1 ... x n n is a vector in V, then
n
T T x j j
j 1
n
x T
j j
j 1
n m
x A
j ij i
j 1 j 1
m n
A x j i .
ij
i 1 j 1
If X is the coordinate matrix of in the ordered basis , then the computation above shows that
AX is the coordinate matrix of the vector T in the ordered basis ’ , because the scalar
n
A x j
ij
j 1
is the entry in the ith row of the column matrix AX. Let us also observe that if A is any m n
matrix over the field F, then
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