Page 70 - Linear Algebra Done Right

Chapter 3. Linear Maps
56
as desired.
The last theorem implies that every finite-dimensional vector space
Because every surjective, it is invertible (see 3.17), and hence V and W are isomorphic,
n
finite-dimensional is isomorphic to some F . Specifically, if V is a finite-dimensional vector
n
vector space is space and dim V = n, then V and F are isomorphic.
n
isomorphic to some F , If (v 1 ,...,v n ) is a basis of V and (w 1 ,...,w m ) is a basis of W, then
why bother with for each T ∈L(V, W), we have a matrix M(T) ∈ Mat(m, n, F). In other
abstract vector spaces? words, once bases have been fixed for V and W, M becomes a function
To answer this from L(V, W) to Mat(m, n, F). Notice that 3.9 and 3.10 show that M is
question, note that an a linear map. This linear map is actually invertible, as we now show.
investigation of F n
would soon lead to 3.19 Proposition: Suppose that (v 1 ,...,v n ) is a basis of V and
vector spaces that do (w 1 ,...,w m ) is a basis of W. Then M is an invertible linear map be-
n
not equal F . For tween L(V, W) and Mat(m, n, F).
example, we would
encounter the null Proof: We have already noted that M is linear, so we need only
space and range of prove that M is injective and surjective (by 3.17). Both are easy. Let’s
linear maps, the set of begin with injectivity. If T ∈L(V, W) and M(T) = 0, then Tv k = 0
matrices Mat(n, n, F), for k = 1,...,n. Because (v 1 ,...,v n ) is a basis of V, this implies that
and the polynomials T = 0. Thus M is injective (by 3.2).
P n (F). Though each of To prove that M is surjective, let
these vector spaces is
 
isomorphic to some a 1,1 ... a 1,n
m
F , thinking of them  . . . . 
A =  . . 
 
that way often adds
a m,1 ... a m,n
complexity but no new
insight. be a matrix in Mat(m, n, F). Let T be the linear map from V to W such
that
m

Tv k = a j,k w j
j=1
for k = 1,...,n. Obviously M(T) equals A, and so the range of M
equals Mat(m, n, F), as desired.

An obvious basis of Mat(m, n, F) consists of those m-by-n matrices
that have 0 in all entries except fora1inone entry. There are mn such
matrices, so the dimension of Mat(m, n, F) equals mn.
Now we can determine the dimension of the vector space of linear
maps from one finite-dimensional vector space to another.