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 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 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

Approach 4: Find a Basis without the Decomposed Polynomial