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Assignment 1

This document contains three math problems. Problem 1 involves vector spaces of polynomials and finding bases of subspaces. Problem 2 involves subspaces of continuous functions related to sine and cosine. Problem 3 involves showing three sets of functions are bases for the same subspace and calculating change of basis matrices.

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21 views1 page

Assignment 1

This document contains three math problems. Problem 1 involves vector spaces of polynomials and finding bases of subspaces. Problem 2 involves subspaces of continuous functions related to sine and cosine. Problem 3 involves showing three sets of functions are bases for the same subspace and calculating change of basis matrices.

Uploaded by

Brady
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Assignment 1

MATH 304 April 8, 2022 Name

Problem 1. Consider the vector space P2 of all polynomials with real coefficients of degree less or equal
than 2 defined on the real line.
We say that p ∈ P2 has a vertex at x0 ∈ R if p(x0 ) ≤ p(x) for all x ∈ R or p(x0 ) ≥ p(x) for all x ∈ R.
(Notice that under this definition a constant polynomial has a vertex at every real number x0 .)

(i) Let s ∈ R be an arbitrary (fixed) number. Let Zs be the set all polynomials p ∈ P2 such that
p(s) = 0, that is, 
Zs = p ∈ P2 : p(s) = 0 .
Prove that Zs is a subspace of P2 . Find a basis of this subspace. What is dim Zs ?

(ii) Let s ∈ R be an arbitrary (fixed) number. Let Vs be the set of all polynomials p ∈ P2 which have
a vertex at s. Prove that Vs is a subspace of P2 . Find a basis of this subspace. What is dim Vs ?

(iii) Let s, u ∈ R be given such that s 6= u. Describe the polynomials in each of the subspaces Zs ∩ Zu ,
Vs ∩ Zu and Vs ∩ Vu . Find a basis for each of these subspaces.

(iv) Let s, u ∈ R be given such that s 6= u. Solve the equation Zs ∩ Zx = Vy ∩ Zu for x and y.

Problem 2. Consider the vector space V of all continuous real valued functions defined on R, see
Example 5 on page 194. The purpose of this exercise is to study some special subspaces of the vector
space V. Let ω be an arbitrary (fixed) real number. Consider the set
n o
Sω := f ∈ V : ∃ a, b ∈ R such that f (t) = a sin(ωt + b) ∀t ∈ R .

(a) Do you see an exceptional value for ω for which the set Sω is particularly simple? Explain.

(b) Prove that Sω is a subspace of V.

(c) For each ω ∈ R find a basis for Sω . Plot the function ω 7→ dim Sω .

Problem 3. Consider the vector space V of all continuous real valued functions defined on R, see
Example 5 on page 194. Consider the following fifteen functions in V

a0 = 1, b0 = 1, c0 = 1,
2
a1 = (cos t) , b1 = cos(2t), c1 = (sin t)2 ,
a2 = (cos t)4 , b2 = cos(4t), c2 = (sin t)4 ,
a3 = (cos t)6 , b3 = cos(6t), c3 = (sin t)6 ,
a4 = (cos t)8 , b4 = cos(8t), c4 = (sin t)8 .

Set   
A = a0 , a1 , a2 , a3 , a4 , B = b0 , b1 , b2 , b3 , b4 , C = c0 , c1 , c2 , c3 , c4 .

(i) Prove that Span A = Span B = Span C.

Denote the common span from the previous item by H. That is H = Span A = Span B = Span C.

(ii) Prove that each set A, B, and C is a basis for H.

(iii) Calculate the change of coordinates matrices:

P , P , P , P , P , P .
B←A C←A A←B C←B A←C B←C

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