Welcome to Math 240! (additions to this web site no longer count towards good deed points)

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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

MondayTuesday is UofT November break

12

Nov 24

HW9

13

Dec 1

Wednesday is a "makeup Monday"! EndofCourse Schedule, Tutorial

F

Dec 8

The Final Exam

Register of Good Deeds

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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 $g(x)\in P_{n1}(R)$ such that $f(x)=(xa)g(x)$. From here, there are several approaches:
Approach 1: Use Isomorphisms
We show that $W$ is isomorphic to $P_{n1}(R)$. Let $B=\{1,x,x^{2},...,x^{n1}\}$ be the standard ordered basis of $P_{n1}(R)$ and $S=\{xa,(xa)x,(xa)x^{2},...,(xa)x^{n1}\}$ be a subset of $W$. Then there is a unique linear transformation $T:P_{n1}\to W$ such that $T(f(x))=(xa)f(x)$ where $f(x)\in B$. Show that $T$ is both onetoone and onto and conclude that $dim(P_{n1})=dim(W)$.
Approach 2: Use the RankNullity Theorem
Let $K=\{1,x,x^{2},...,x^{n1},x^{n}\}$ be the standard ordered basis of $P_{n}$ 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 $T:P_{n}(R)\to R$ by $T(\displaystyle \sum _{i=1}^{n}c_{i}g_{i}(x))=\displaystyle \sum _{i=1}^{n}c_{i}g_{i}(a)$. Then it is easy to show that $T$ is both welldefined and linear. Afterwards, show that $rank(T)=1$ and use the ranknullity theorem to conclude that $dim(W)=n$.
Approach 3: Find a Basis with the Decomposed Polynomial
This approach is straightforward. Show that $S=\{xa,(xa)x,(xa)x^{2},...,(xa)x^{n1}\}$ 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: $S=\{xa,(x^{2}a^{2}),(x^{3}a^{3}),...,(x^{n}a^{n})\}$.
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.
Nikita