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Solved–Homework 6 –Solution

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In each of the following you are given a statement, which may be true or false. Determine whether the statement is correct and show how you reached this conclusion. ( ) ( )( )( ) i. 1 2 2 spanf 2 0 ; 1 1 ; 2 1 g 2 1 1 1 3 0…

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  1. In each of the following you are given a statement, which may be true or false. Determine whether the statement is correct and show how you reached

this conclusion.

( ) ( )( )( )

i.

1

2

2 spanf

2

0

;

1

1

;

2

1

g

2

1

1

1

3

0

2

1

ii. 2 + 3x + 2x2

x3 2 spanf1

x3; 2 + x + x2; 3

xg

);

(

); (

iii. spanf(

5

3

); (

4

1

)g spanf(

1

1

3

0

0

5

2

1

1

1

0

1;

0

2

0

1

1

1

iv. span

0

1

;

0

1

= span

2

0

1

;

0

2

1

1

3

1

1

f@ 1 A @ 0 Ag

f@ 1 A @ 1 A @

1 Ag

  • ) ( )

v. f

1

;

1

is a spanning set for R2

.

1

2

  • ) ( )

vi. f

1

;

1

is a spanning set for R2

.

1

1

vii. f1

x + x2; x

x2 + x3; 1 + x2

x3; x3g is a spanning set for R3[x].

vii. f(

1

1

); (

2

1

); (

0

1

)g is a spanning set for M2(R).

1

1

1

0

3

1

)

2 1

2 1 g

  1. In each of the following you are given a vector space (you do not need to prove that this is indeed a vector space). Find a spanning set for each of these vector spaces.

    1. fA 2 Mn(R) : A is diagonal g (The de nition of a diagonal matrix was

given in HW2).

0

a + b + c

1

: a; b; c

R

ii.

a

2b

f

B

34c

b

C

2

g

B

C

@

a

2c

A

(

)

(

)g

iii.

fA 2 M2

(R):A

2

=

0

1

0

  1. fp(x) 2 R3[x] : p′ (1) = 0g

  1. fp(x) 2 Rn[x] : p(1) = p( 1)g

2

3. In each of the following you are given a set, determine whether it is linearly

independent or linearly dependent, show how you reach your conclusion.

i. f(

1

1 ) (

3

0

) (

2

1 )g

0

2

0

1

1

1

2

1

3

1 0

1

1 0

1

  1. f@ 2 A; @ 0 A; @ 2 Ag

1 1 1

iii. f1 x3; 2 + x + x2; 3 x; 1 + x + x2 + x3g

    1. ff(x) = sin2 x; g(x) = cos2(x); h(x) = 1g (Note that h(x) is the constant function which is equal to 1 for every x).

  1. Let V be a vector space and w1; w2; w3 in V be such that fw1; w2; w3g is linearly independent. Prove or disprove the following claims.

    1. The set fw1 + w2 + w3; w2 + w3; w3g is linearly independent.

    1. The set fw1 + 2w2 + w3; w2 + w3; w1 + w2g is linearly independent.

  1. Let V be a vector space and let S V and T V be two nite subsets of V . Prove or disprove the following claims.

    1. If S T and S is linearly independent then T is linearly independent.

    1. If S T and T is linearly independent then S is linearly independent.

    1. If S and T are linearly independent then S\T is either empty or linearly independent (Remark: sometimes people consider an empty set to be linearly independent).

    1. If S and T are linearly independent then S [ T is linearly independent.

    1. If W = spanS and U = spanT then W + U = span(S [ T ). (The sum of two subspaces was de ned in HW4).

  1. Let v1; :::; vn 2 Rm. Prove or disprove the following claims. (Hint: several of these were given in previous HW’s, or in class during our studies of Chapters 1 and 2, but with di erent formulations. As usual, you are welcome to use whatever was proved in class without repeating the proof.)

    1. If fv1; :::; vng is linearly independent then n m.

    1. If n m then fv1; :::; vng is linearly independent.

    1. If fv1; :::; vng spans Rm then n m.

    1. If n m then fv1; :::; vng spans Rm

    1. If fv1; :::; vng is both linearly independent and spans Rm then n = m.

    1. If n = m then: fv1; :::; vng is linearly independent i fv1; :::; vng spans

Rm.