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
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.
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
fp(x) 2 R3[x] : p′ (1) = 0g
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
f@ 2 A; @ 0 A; @ 2 Ag
1 1 1
iii. f1 x3; 2 + x + x2; 3 x; 1 + x + x2 + x3g
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).
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.
The set fw1 + w2 + w3; w2 + w3; w3g is linearly independent.
The set fw1 + 2w2 + w3; w2 + w3; w1 + w2g is linearly independent.
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.
If S T and S is linearly independent then T is linearly independent.
If S T and T is linearly independent then S is linearly independent.
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).
If S and T are linearly independent then S [ T is linearly independent.
If W = spanS and U = spanT then W + U = span(S [ T ). (The sum of two subspaces was de ned in HW4).
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.)
If fv1; :::; vng is linearly independent then n m.
If n m then fv1; :::; vng is linearly independent.
If fv1; :::; vng spans Rm then n m.
If n m then fv1; :::; vng spans Rm
If fv1; :::; vng is both linearly independent and spans Rm then n = m.
If n = m then: fv1; :::; vng is linearly independent i fv1; :::; vng spans
Rm.
