Solved-Homework 5- Solution

Description

1. 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.

1. 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 R}2 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

2

_{vi. The set R}2 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	:

2. 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).

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

2. 2. For every scalar 2 R we have 0_V = 0_{V .}

2. 3. 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

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

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

0 1

x₁

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

3

6. Let A 2 M_{3 4}(R) be a speci c matrix. in this question V = M₄(R) and

W = fB 2 M₄(R) : AB = 0g.

8. 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

.

4. 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.

4. 1. U \ W is also a subspace of V .

4. 2. U [ W is also a subspace of V .

4. 3. 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 .

5. _{i. Give an example of a subset of R}2 that is closed under scalar multipli-

cation but not addition.

2. _{Give an example of a subset of R}2 that is closed under addition but not scalar multiplication.

3. Give an example of a subset of R^{2 that is closed under neither.}

4. _{Identify all of the subspaces of R}3, you do not need to prove your claim, just provide a ‘good guess’.