MATH 221, Spring 2025

Problem Set 9
April 11, 2024

This assignment is associated with Chapter 6 of your textbook. These sections discuss general
vector spaces, including the ten properties that must be fulfilled for a set to be considered a
vector space. The theory surrounding subspaces, spanning sets, linear independence, and dimension
parallel (and generalize) those we saw in Chapter 5, which discussed the specific vector space Rn .
Section 6.4 introduces the idea of infinite dimensional vector spaces.

For all of the following problems, be sure to show all of your work to demonstrate how you have
arrived at your solution.

1. Determine which of the following sets are vector spaces under the given operations. For those
that are not vector spaces, list the axioms that fail to hold.

(a) The set of all triples of real numbers (x, y, z) with the operations

(x, y, z) + (x0 , y 0 , z 0 ) = (x + x0 , y + y 0 , z + z 0 ) and k(x, y, z) = (0, 0, 0)

(b) The set of all real-valued functions f defined everywhere on the real line and such that
f (1) = 0 with the operations

(f + g)(x) = f (x) + g(x) and (kf )(x) = kf (x)

(c) The set of all 2 ⇥ 2 matrices of the form


a a+b
a+b b

with the standard matrix addition and scalar multiplication.

2. Determine whether the following are subspaces of R2 , R2⇥2 , and C[a, b] (continuous functions
defined on the interval [a, b]), respectively.
⇢
x1
(a) U = : x1 = x2
x2
(b) all 2 ⇥ 2 matrices such that det(A) = 0
(c) all f 2 C[a, b] such that
Z b
f (x)dx = 0
a

1
3. Show that the solution vectors of a consistent nonhomogeneous system of m linear equations
in n unknowns do not form a subspace of Rn . (Hint: A solution vector, xi , satisfies Axi = b.)

4. Let f (x) = cos2 x and g(x) = sin2 x. Which of the following lie in the space spanned by
f (x) and g(x) (Hint: Which of the following can be written as a linear combination of these
“vectors”?

(a) cos 2x
(b) 3 + x2
(c) 1
(d) sin x
(e) 0

5. Show that f1 (x) = 1, f2 (x) = ex , and f3 (x) = e2x form a linearly independent set of vectors
in C 2 ( 1, 1), i.e. the space of real-valued functions with continuous 2nd derivatives on the
interval ( 1, 1).

6. By inspection, explain why the following sets of vectors are not bases for the indicated vector
spaces

(a) u1 = (1, 2), u2 = (0, 3), u3 = (2, 7) for R2

(b) u1 = ( 1, 3, 2), u2 = (6, 1, 1) for R3
(c) p1 = 1 + x + x2 , p2 = x 1 for P2 , i.e. the space of quadratic polynomials
(d)     
1 1 6 0 3 0 5 1 7 1
A= ,B = ,C = ,D = ,E =
2 3 1 4 1 7 4 2 2 9
for R2⇥2

7. Find a basis for the subspace of P2 spanned by the given vectors. What is the dimension of
the subspace? (Hint: See example 6.3.1. Consider the matrix A that will result from finding
the coefficients in a linear combination of the polynomials and recall that reduction to REF
can help us count solutions and free variables, and determine linear independence.)

(a) 1+x 2x2 , 3 + 3x + 6x2 , 9

(b) 1 + x, x2 , 2 + 2x2 , 3x
(c) 1 + x 3x2 , 2 + 2x 6x2 , 3 + 3x 9x2

2
 1= k(x y z) (kX ky kz) + (x y z)
1)a)
, ,
=
Fuils Axiom , ,
, ,

6) If flu) & rector all axions satisfied

o
g(1) 0 : :
space
= ,
=

( + g)() =
f(i) + g()) = 0+ 0 = 0 -

kf()) -
(kf)(1) = = k 0 = 0
.

c) Define matrix as M(a b) . M, =

(a , , bi) Mz =
(ac , b2)
(atbis (az b)

[atan J
+ +
M , t Mc =
=
M(a , + a) ,
bi + b)
(a +bi)+ (az + bi)
,
bitb2

<M
(4 clatby Maa ,
=
=

under addition & multiplication

: rector space , closed

2)a) v =

q(*] : x=
x]"utE = x, +x

X=1 Xu I + 1 = 2
yes
=

: =

3 .
XXIEU
[XF@X2 > 2x, = 222 v
4 2 =
x =

def(B)
[337 deta)
= 0
b) A B
/4471
Take :
,
: 0 and =

A+ B
[ii] / / ] (3) +
2 .
= =

42 = -12 (violated
det (A+ B) 30-
= 0

: no

c)
/Max = 0 take f(x) = 1.
Scoldx = 0 fIX)tOr

2 .

f(x) = 0 g(x) = 0

:
yes
f(x) + g(x) ((0)
=

+ (0)dx = o

< f(x)
3 =

2) (0)dx = 2 .

0 = or

3) Ax = b ,
b0

1. since d o
-

> solut (violated)

Take A =
[1 1] b =
93]
*
X =
[ 3t] tEI

[11][ x2] 3
=

t =o
if
x, + xz = 3 Xz = t
30 (violated)
[3-07
x =
=

X1 = 3 -

Xz =
3 -
E

4) (os2X + sin2x = 1
a) cos2X = 1032X-sin2x =
f(x)-g(x) = in
span-
: f(x) +
g(x) = 1
x2
b) 3+ not in
span
X

f(x) = 1-sinty polynomial

unrelated to sinx/cost
g(x) = 1 -

((X() 1 = cos2x + sin2x =

f(x) + g(x) = in span-
X
d) sinx not in span
sin and cos2X
can't be written as a combo of
e) 0 = 0 .

f(x) + o .

g(x) = in span-
5) a, fi(x) + acfz(x) + asts(x) = 0 for all xt (-00 90) ,

prove :

x= 0

x= 1
,
a ,

h + Aze
= a) =
ay
+
=

aze"
0

= 0
[,e e = Vandermoude Marya

(e- 1) (e 1) (e = e) to
, =

0
X=2 , di + aze" + azei = : vectors are independent

it's X
linearly dependent > can't be basis
=

6) a) I
has 3 rectors so
is 2-dimensional , yet this set

X
13 is 3-dimensional 2 rectors can't span that space
b) ,

which X
c) P2 is a 3-dimensional space ,
in this set ,
there's
only 2
polynomials is not
enough .

2*

d) has 4 dimensions ,
but the set contains 5 matrices - > linearly dependent =

> can't be basis X

7)a) Pz = 3 dim

&
1-3-)ERIT On 6

Basis
G + +X 2x2 3 + 3x+ 6x2 ,
974
-

[
: ,

Dimension :
3
R3/-18

-ERR
I

O I
1 I
·

0
i
I

b C

b)

S
I
- 1020

BasGxx a
Dimension : 3 "I

02

[v]
2
0
-
 a

I By c
.

4)

Basis
G1 + X-3x2]
,
1000]
:

Dimension :
11