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 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 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 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} 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 $T(f(x))=(x-a)f(x)$ where 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 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 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 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 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 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 $dim(W)=n$.

Approach 3: Find a Basis with the Decomposed Polynomial

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

Approach 4: Find a Basis without the Decomposed Polynomial

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

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.