Page 55 - Linear Algebra Done Right

Null Spaces and Ranges
write ST instead of S ◦ T. You should verify that ST is indeed a linear
map from U to W whenever T ∈L(U, V) and S ∈L(V, W). Note that
ST is defined only when T maps into the domain of S. We often call 41
ST the product of S and T. You should verify that it has most of the
usual properties expected of a product:
associativity
(T 1 T 2 )T 3 = T 1 (T 2 T 3 ) whenever T 1 , T 2 , and T 3 are linear maps such
that the products make sense (meaning that T 3 must map into the
domain of T 2 , and T 2 must map into the domain of T 1 ).

identity
TI = T and IT = T whenever T ∈L(V, W) (note that in the first
equation I is the identity map on V, and in the second equation I
is the identity map on W).

distributive properties
(S 1 + S 2 )T = S 1 T + S 2 T and S(T 1 + T 2 ) = ST 1 + ST 2 whenever
T, T 1 ,T 2 ∈L(U, V) and S, S 1 ,S 2 ∈L(V, W).

Multiplication of linear maps is not commutative. In other words, it
is not necessarily true that ST = TS, even if both sides of the equation
make sense. For example, if T ∈L(P(R), P(R)) is the differentiation
map defined earlier in this section and S ∈L(P(R), P(R)) is the mul-
2
tiplication by x map defined earlier in this section, then
2
2
((ST)p)(x) = x p (x) but ((TS)p)(x) = x p (x) + 2xp(x).

2
In other words, multiplying by x and then differentiating is not the
2
same as differentiating and then multiplying by x .
Null Spaces and Ranges

For T ∈L(V, W), the null space of T, denoted null T, is the subset Some mathematicians
of V consisting of those vectors that T maps to 0: use the term kernel
instead of null space.
null T ={v ∈ V : Tv = 0}.

Let’s look at a few examples from the previous section. In the dif-
ferentiation example, we defined T ∈L(P(R), P(R)) by Tp = p . The