Question 26 on Page 57 in Homework 5
Let and be a subspace of . Find .
First, let . Then we can decompose since there is a such that . From here, there are several approaches:
Approach 1: Use Isomorphisms
We show that is isomorphic to . Let be the standard ordered basis of and be a subset of . Then there is a unique linear transformation such that where . Show that is both one-to-one and onto and conclude that .
Approach 2: Use the Rank-Nullity Theorem
Let be the standard ordered basis of and . Then where and . Define by . Then it is easy to show that is both well-defined and linear. Afterwards, show that and use the rank-nullity theorem to conclude that .
Approach 3: Find a Basis with the Decomposed Polynomial
This approach is straightforward. Show that is a basis of .
Approach 4: Find a Basis without the Decomposed Polynomial
This approach requires a little more cleverness when constructing the basis: .
Boris's Section Only
If you use in your proof Corollary 1 of the Fundamental Theorem of Algebra, then please cite it as "Corollary 1 of the Fundamental Theorem of Algebra". Do not cite it as the "Fundamental Theorem of Algebra" since that means you are citing the fundamental theorem instead of its corollary.