14-240/Tutorial-November4

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Boris

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 $f(x)=(x-a)g(x)$. From here, there are several approaches:

Approach 1: Use Isomorphisms

We show that $W$ is isomorphic to $P_{n-1}(R)$. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = \{1, x, x^2, ..., x^{n - 1}\}} be the standard ordered basis of $P_{n-1}(R)$ and $S=\{x-a,(x-a)x,(x-a)x^{2},...,(x-a)x^{n-1}\}$ be a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} . Then there is a unique linear transformation $T:P_{n-1}\to W$ such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(f(x)) = (x - a)f(x)} where $f(x)\in B$. Show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is both one-to-one and onto and conclude that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dim(P_{n - 1}) = dim(W)} .

Approach 2: Use the Rank-Nullity Theorem

Let $K=\{1,x,x^{2},...,x^{n-1},x^{n}\}$ be the standard ordered basis of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_n} and $f(x)\in P_{n}(R)$. Then $f(x)=\displaystyle \sum _{i=1}^{n}c_{i}g_{i}(x)$ where $c_{i}\in R$ and $g_{i}(x)\in K$. Define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T: P_{n}(R) \to R} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\displaystyle\sum_{i=1}^{n} c_ig_i(x))= \displaystyle\sum_{i=1}^{n} c_ig_i(a)} . Then it is easy to show that $T$ is both well-defined and linear. Afterwards, show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle rank(T) = 1} and use the rank-nullity theorem to conclude that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dim(W) = n} .

Approach 3: Find a Basis with the Decomposed Polynomial

This approach is straightforward. Show that $S=\{x-a,(x-a)x,(x-a)x^{2},...,(x-a)x^{n-1}\}$ is a basis of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} .

Approach 4: Find a Basis without the Decomposed Polynomial

This approach requires a little more cleverness when constructing the basis: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.