MATH 4340, Section 1, Homework Assignment 2

| August 31, 2017

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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