Homework 5 Solution

In each of the following you are given a set and two operations: A 'sum',



acting between two elements in the set, and a 'multiplication by scalar', acting between one element in the set and a scalar from R. In each case determine whether the set with these two operations gives a vector space over R. If it is a vector space then prove this fact. If it is not a vector space then show this by giving a counterexample. In this question you are allowed to use only the de nition of a vector space, not any other claim given in class.




The set P2(R) with the usual operations of summation and multiplica-tion by scalar de ned for polynomials.



ii. The set
0
x
1








B
C






f
y
: x
y + 2z = 0
g
z
w






B


C








@


A






with the usual operations of summation and multiplication by scalar de ned for n-tuples.

iii. The set R2 with the operations

( x2
)


( y2
) =
(
0
)
x1




y1


x1 + y1


and


( x2
)
= (
0
):


⊙








x1




x1







iv. The set R2 with the operations (note the locations of y2 in the de nition)
( x2
)
( y2
) =
( x2
+ y2
)
x1


y1


x1
+ y2


and


⊙ (


) = (




):






x1
x1




x2
x2


v. The set R2 with the operations






2 )
( x2
)
( y2
)
= (
x2
+ y2
x1




y1




x1
+ y1
3
and


















⊙ (
x1


(
x1
3 + 3
):
x2 ) =
x2
2 + 2
1






2




vi. The set R2 with the operations






(
x1


(
y1


x1 + y1
)


x2 )
y2 ) = ( x2 + y2
and


⊙ (




















x1






2 x1






(
x2
) = ( 2 x2 ):






)


















vii. The set f
x1
2 R : x1
0; x2
0g with the operations
x2


( x2
) ( y2
)
= ( x2y2
)




x1






y1




x1y1




and




⊙ ( x2
)


( x2
)
























x1


=
x1
:































Let V be a vector space over R. Prove the following claims. (That is, prove that each one of these claims follows from the de nition of a vector space).



For every v 2 V we have 2v + v = 3v.



For every scalar 2 R we have 0V = 0V .



The additive inverse of the additive inverse of a vector is equal to the
vector, that is, if v 2 V then ( v) = v.




iv. For every u; v; w; z 2 V we have (u + w) + (v + z) = w + (u + (v + z)).




3. In each of the following you are given a vector space V and a subset W of




this space. Determine whether the subset is a subspace. If you claim that the answer is 'yes' then prove this. If you claim that the answer is no then show this by providing a counterexample.

0 1









x1




















i. V = R4
and W =


B
x2
C
: x1


0; x2


0; x3


0; x4
0 .
f
B
x4
C




















g






@
x3
A














































ii. V = M2(R) and W = f(
x
x


3y
) : x; y 2 Rg.


y
2x
+
y


V = R4[x] and W = fp(x) 2 R4[x] : p(1) = 1g.



V = R4[x] and W = fp(x) 2 R4[x] : p(1) = 0g.
0 1

x1

V = R3 and W = f@ x2 A : x1 2 Q; x2 2 Q; x3 2 Qg. x3






3




Let A 2 M3 4(R) be a speci c matrix. in this question V = M4(R) and



W = fB 2 M4(R) : AB = 0g.




V = ff : R 7!Rg and



W = ff 2 V : f is twice di arentiable and f′′(x) + 3f′(x) f(x) = 0 8x 2 Rg




.




Let V be a vector space over R and let W V and U V be two subspaces of V . The following claims are either true or false. Determine whether they are true or false and prove or disprove using a counterexample accordingly.



U \ W is also a subspace of V .



U [ W is also a subspace of V .



We de ne the following subset of V :



U + W := fu + w : u 2 U; w 2 W g:




In this part of the question the claim is: U + W is a subspace of V .

i. Give an example of a subset of R2 that is closed under scalar multipli-



cation but not addition.

Give an example of a subset of R2 that is closed under addition but not scalar multiplication.
Give an example of a subset of R2 that is closed under neither.
Identify all of the subspaces of R3, you do not need to prove your claim, just provide a 'good guess'.