Page 130 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes
n m n
T x A x
j j ij j i ...(2)
j 1 i 1 j 1

defines a linear transformation T from V into W, the matrix of which is A, relative to  , '.
Theorem 1: Let V be an n-dimensional vector space over the field F and W an m-dimensional
vector space over F. Let  be an ordered basis for V and ' an ordered basis for W. For each
linear transformation T from V into W, there is an m×n matrix A with entries in F such that
T  '=A 

for every vector in V. Furthermore, T A is a one-one correspondence between the set of all
linear transformations from V into W and the set of all m×n matrices over the field F.

The matrix A which is associated with T in Theorem 1 is called the matrix of T relative to the
ordered basis , '.  Note that Equation (1) says that A is the matrix whose columns A ,...,A are
1 n
given by

A j T j  ', j = 1, ..., n.
If U is another linear transformation from V into W and B B 1 ,...,B is the matrix of U relative to
n
the ordered basis , '  then cA B is the matrix of cT U relative , '.  That is clear because

cA B c T '  U ' 
j j j j
cT j U j ' 

cT U j  '.

Theorem 2: Let V be an n-dimensional vector space over the field F and let W be an m-dimensional
vector space over F. For each pair of ordered bases , '  for V and W respectively, the function
which assigns to a linear transformation T its matrix relative to , '  is an isomorphism between
the space L(V,W) onto the set of m×n matrices over the field F.

Proof: We observed above that the function in question is linear, and as stated in Theorem 1, this
function is one-one and maps L(V, W) onto the set of m×n matrices.

We shall be particularly interested in the representation by matrices of linear transformations
of a space into itself, i.e., linear operators on a space V. In this case it is most convenient to use the
same ordered basis in each case, that is, to take = '  . We shall then call the representing matrix
simply the matrix of T relative to the ordered basis . Since this concept will be so important to
us, we shall review its definition. If T is a linear operator on the finite-dimensional vector space
V and = ,..., is an ordered basis for V, the matrix of T relative to  (or, the matrix of T in
1 n
the ordered basis ) is the n×n matrix A whose entries A are defined by the equations
ij
n
T A , j = 1, ..., n ...(3)
j ij i
i 1
One must always remember that this matrix representing T depends upon the ordered basis  ,
and that there is a representing matrix for T in each ordered basis for V. (For transformations of

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