14-240/Tutorial-November4: Difference between revisions

Revision as of 17:21, 29 November 2014

#	Week of...	Notes and Links
Welcome to Math 240! (additions to this web site no longer count towards good deed points)
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
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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 that

W

is isomorphic to

P_{n-1}

. 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)

where

f(x)\in P_{n-1}

. Then show that

T

is one-to-one and onto.

Approach 2: Use the Rank-Nullity Theorem

Approach 3: Find a Basis with the Decomposed Polynomial

Approach 4: Find a Basis without the Decomposed Polynomial

@@ Line 8: / Line 8: @@
+:First, let <math>f(x) \in W</math>.  Then we can decompose <math>f(x)</math> since there is a <math>g(x) \in P_{n - 1}(R)</math> such that <math>f(x) = (x - a)g(x)</math>.  From here, there are several approaches:
-There are several approaches to this problem.
 '''Approach 1: Use Isomorphisms'''
+:We that <math>W</math> is isomorphic to <math>P_{n - 1}</math>.  Let <math>B = \{1, x, x^2, ..., x^{n - 1}\}</math> be the standard ordered basis of <math>P_{n - 1}(R)</math> and <math>S = \{x - a, (x - a)x, (x - a)x^2, ..., (x - a)x^{n - 1}\}</math> be a subset of W.  Then there is a unique linear transformation <math>T:P_{n - 1} \to W</math> such that <math>T(f(x)) = (x - a)</math> where <math>f(x) \in P_{n - 1}</math>.  Then show that <math>T</math> is one-to-one and onto.
 '''Approach 2: Use the Rank-Nullity Theorem'''
-'''Approach 3: Find a Basis by Decomposing the Polynomial'''
+'''Approach 3: Find a Basis with the Decomposed Polynomial'''
-'''Approach 4: Find a Basis without Decomposing the Polynomial'''
+'''Approach 4: Find a Basis without the Decomposed Polynomial'''