Direct Sum of Linear Transformations and Direct Sum of Matrices (Pt. II)
Point of post: This is a literal continuation of this post. Treat it as one contiguous object.
We may make a similar statement about the kernels of the direct sum of linear transformations. Namely:
Theorem: Let , and be as in the definition of direct sum. Then,
Proof: We prove first that
To see this we first that if where of course then
So that . Conversely we see that if then
but since and we may conclude (from a common characterization of independence of subspaces) that or that or that from where it follows that from where it indeed follows that . Thus, to prove that it suffices to notice (similarly to the proof of the last theorem) that for each we have that
From these two we can prove the following theorem:
Theorem: Let , and be as in the definition of direct sum. Then, if and only if is an isomorphism.
Proof: Suppose first that is an isomorphism for each , then we note that by the above two theorems that
and thus as required.
Conversely, suppose that then we see from our two previous theorems that
from where it clearly follows (recalling that ) that . Similarly, we have that
from where it follows that so that and so . Thus, combining these two gives that is an isomorphism for each .
1. Halmos, Paul R. Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print