14-240/Tutorial-November4

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Question 26 on Page 57 in Homework 5

Let $a\in R$ and $W=\{f\in P_{n}(R):f(a)=0\}$ be a subspace of $P_{n}(R)$. Find $dim(W)$.

First, let $f(x)\in W$. Then we can decompose $f(x)$ since there is a $g(x)\in P_{n-1}(R)$ such that $f(x)=(x-a)g(x)$. Let $B=\{1,x,x^{2},...,x^{n-1}\}$ be the standard ordered basis of $P_{n-1}(R)$. From here, there are several approaches:

Approach 1: Use Isomorphisms

We show that $W$ is isomorphic to $P_{n-1}(R)$. Let $S=\{x-a,(x-a)x,(x-a)x^{2},...,(x-a)x^{n-1}\}$ be a subset of $W$. Then there is a unique linear transformation $T:P_{n-1}\to W$ such that $T(f(x))=(x-a)f(x)$ where $f(x)\in B$. Show that $T$ is one-to-one and onto and conclude that $dim(P_{n-1})=dim(W)$.

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