Your cart is currently empty!
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…
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.