14-240/Tutorial-November4

DBN: Classes: 2014-15: 14-240 / Navigation Welcome to Math 240! (additions to this web site no longer count towards good deed points) # Week of... Notes and Links 1 Sep 8 About This Class, What is this class about? (PDF, HTML), Monday, Wednesday 2 Sep 15 HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf 3 Sep 22 HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf 4 Sep 29 HW3, Wednesday, Tutorial, HW3_solutions.pdf 5 Oct 6 HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf 6 Oct 13 No Monday class (Thanksgiving), Wednesday, Tutorial 7 Oct 20 HW5, Term Test at tutorials on Tuesday, Wednesday 8 Oct 27 HW6, Monday, Why LinAlg?, Wednesday, Tutorial 9 Nov 3 Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial 10 Nov 10 HW8, Monday, Tutorial 11 Nov 17 Monday-Tuesday is UofT November break 12 Nov 24 HW9 13 Dec 1 Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial F Dec 8 The Final Exam Register of Good Deeds Add your name / see who's in!

Boris

Question 26 on Page 57 in Homework 5

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 a \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 W = \{f \in P_n(R): f(a) = 0\}} be a subspace 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(R)} . Find 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)} .

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 $B=\{1,x,x^{2},...,x^{n-1}\}$ 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 $W$. 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 both one-to-one and onto and conclude that $dim(P_{n-1})=dim(W)$.

Approach 2: Use the Rank-Nullity Theorem

Let $K=\{1,x,x^{2},...,x^{n-1},x^{n}\}$ be the standard ordered basis of $P_{n}$ 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 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) = \displaystyle\sum_{i=1}^{n} c_ig_i(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 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 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 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.