# MATH 4340, Section 1, Homework Assignment 2

Question

MATH 4340, Section 1, Homework Assignment 2

Understanding Vector Spaces

Due: 11:15 am, Monday, January 25, 2016

Point value: 20

The purpose of this homework assignment is to help you

1. verify properties of vector spaces;

2. ﬁnd or validate spanning sets and bases of vector spaces;

3. use concepts associated with linear independence of vectors.

Each individual is expected to complete his or her own assignment. Show all work and justify all

conclusions. No late homework will be accepted.

1. (3 pts.) Let S = C 1 [0, π] := {f : f and f are continuous on [0, π]} .

(a) Give two vectors in S. Show why these vectors are in S.

(b) Is it true that for f (x) ∈ S, we have αf (x) ∈ S for α ∈ I Why or why not?

R?

(c) Show that S is closed under vector addition and scalar multiplication. You must use

theorems on continuity from single variable calculus to show these properties.

2x − y

2. (3 pts.) Let S = I 3 , W = y + 2 : x, y ∈ I .

R

R

x + 5y

(a) Give two vectors in W .

(b) Is W a subspace? That is, is W closed under vector addition and scalar multiplication?

Justify your answer.

3. (6 pts.) Let P3 := ax3 + bx2 + cx + d : a, b, c, d ∈ I , the set of all cubic functions with

R

coeﬃcients in the reals.

(a) Are elements of P3 (x) in S from Problem 1 above? Why or why not?

(b) Show that P3 (x) is closed under vector addition and scalar multiplication. Since P3 (x)

is a subset of the vector space of continuous functions, showing these properties actually

veriﬁes that P3 (x) is a subspace.

(c) Give an example of a function in S Problem 1 above but not in P3 .

(d) Show that any function in P3 can be represented as a linear combination of the functions

in the set W := 1 + x, x2 , x2 − 2x, x3 . That is, you need to show you can ﬁnd c1 , c2 ,

c3 and c4 such that

c1 (1 + x) + c2 x2 + c3 x2 − 2x + c4 x3 = ax3 + bx2 + cx + d

for any real numbers a, b, c, and d.

(e) Are the functions in W linearly independent? To answer this question, consider the

answer above in (c) when a = b = c = d = 0.

1

4. (5 pts.) Let

V := f (x) ∈ C 2 (0, 1) : x2 f (x) + cos (x) f (x) − xf (x) = 0, 0 < x < 1; f (0) = 0, f (1) = 0 .

Note: C 2 (0, 1) is the set of continuous functions with ﬁrst and second continuous derivatives

deﬁned on the open interval (0, 1).

(a) Show that V is a subspace of C 2 (0, 1). That is, show that the set V is closed under

vector addition and scalar multiplication. If a function (or vector) g(x) is an element of

V , then g satisﬁes the diﬀerential equation and the boundary conditions. You are not

required to solve the diﬀerential equation in order to answer this question.

(b) Modify V such that it is no longer a subspace of C 2 (0, 1). Show why your modiﬁed set

does not satisfy the properties associated with subspaces.

5. (3 pts.) Consider the set of vectors V = {v1 , v2 , v3 } in I 3 , where

R

−1

2

1

v2 = 1 ,

v3 = 0 .

v1 = 4 ,

−1

2

−1

(a) Show the vectors in V are linearly independent.

(b) Consider the vector

−3

w = 1 .

1

Find c1 , c2 , and c3 such that

c1 v1 + c2 v2 + c3 v3 = w.

2

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