# 14-240/Tutorial-November4

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

## Boris

#### Question 26 on Page 57 in Homework 5

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

First, let ${\displaystyle f(x)\in W}$. Then we can decompose ${\displaystyle f(x)}$ since there is a ${\displaystyle g(x)\in P_{n-1}(R)}$ such that ${\displaystyle f(x)=(x-a)g(x)}$. From here, there are several approaches:

Approach 1: Use Isomorphisms

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

Approach 2: Use the Rank-Nullity Theorem

Let ${\displaystyle K=\{1,x,x^{2},...,x^{n-1},x^{n}\}}$ be the standard ordered basis of ${\displaystyle P_{n}}$ and ${\displaystyle f(x)\in P_{n}(R)}$. Then ${\displaystyle f(x)=\displaystyle \sum _{i=1}^{n}c_{i}g_{i}(x)}$ where ${\displaystyle c_{i}\in R}$ and ${\displaystyle g_{i}(x)\in K}$. Define ${\displaystyle T:P_{n}(R)\to R}$ by ${\displaystyle T(\displaystyle \sum _{i=1}^{n}c_{i}g_{i}(x))=\displaystyle \sum _{i=1}^{n}c_{i}g_{i}(a)}$. Then it is easy to show that ${\displaystyle T}$ is both well-defined and linear. Afterwards, show that ${\displaystyle rank(T)=1}$ and use the rank-nullity theorem to conclude that ${\displaystyle dim(W)=n}$.

Approach 3: Find a Basis with the Decomposed Polynomial

This approach is straightforward. Show that ${\displaystyle S=\{x-a,(x-a)x,(x-a)x^{2},...,(x-a)x^{n-1}\}}$ is a basis of ${\displaystyle W}$.

Approach 4: Find a Basis without the Decomposed Polynomial

This approach requires a little more cleverness when constructing the basis: ${\displaystyle S=\{x-a,(x^{2}-a^{2}),(x^{3}-a^{3}),...,(x^{n}-a^{n})\}}$.

#### Cite Carefully

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 Do not cite it as the "Fundamental Theorem of Algebra" since that means you are citing the fundamental theorem instead of its corollary.