Page 62 - Linear Algebra Done Right

Chapter 3. Linear Maps
48
The Matrix of a Linear Map
We have seen that if (v 1 ,...,v n ) is a basis of V and T : V → W is
linear, then the values of Tv 1 ,...,Tv n determine the values of T on
arbitrary vectors in V. In this section we will see how matrices are used
as an efficient method of recording the values of the Tv j ’s in terms of
a basis of W.
Let m and n denote positive integers. An m-by-n matrix is a rect-
angular array with m rows and n columns that looks like this:

 
a 1,1 ... a 1,n
 . . 
3.7  . . . .  .
 
a m,1 ... a m,n
Note that the first index refers to the row number and the second in-
dex refers to the column number. Thus a 3,2 refers to the entry in the
third row, second column of the matrix above. We will usually consider
matrices whose entries are elements of F.
Let T ∈L(V, W). Suppose that (v 1 ,...,v n ) is a basis of V and
(w 1 ,...,w m ) is a basis of W. For each k = 1,...,n, we can write Tv k
uniquely as a linear combination of the w’s:

3.8 Tv k = a 1,k w 1 + ··· + a m,k w m ,

where a j,k ∈ F for j = 1,...,m. The scalars a j,k completely determine
the linear map T because a linear map is determined by its values on
a basis. The m-by-n matrix 3.7 formed by the a’s is called the matrix
of T with respect to the bases (v 1 ,...,v n ) and (w 1 ,...,w m ); we denote
it by

M T, (v 1 ,...,v n ), (w 1 ,...,w m ) .
If the bases (v 1 ,...,v n ) and (w 1 ,...,w m ) are clear from the context
(for example, if only one set of bases is in sight), we write just M(T)

instead of M T, (v 1 ,...,v n ), (w 1 ,...,w m ) .
As an aid to remembering how M(T) is constructed from T, you
might write the basis vectors v 1 ,...,v n for the domain across the top
and the basis vectors w 1 ,...,w m for the target space along the left, as
follows: