14-240/Tutorial-November4

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Boris

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 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 . Then show that is one-to-one and onto.


Approach 2: Use the Rank-Nullity Theorem

Approach 3: Find a Basis with the Decomposed Polynomial

Approach 4: Find a Basis without the Decomposed Polynomial