Homework 8 Solution

Find the dimensions of each one of the spaces in HW7 Question 1.



Find a basis to the following spaces and determine the dimension of each of
these spaces.
1
1
;
0
4
1 ;
0
3
1






i. span


0
.










1






2








1












f@ 2 A @ 1 A @
1 Ag






ii. span


0
1
1
;
0
1
1 ;
0


1
1


.


f
B
1
C B
1


C B


5
C
g




2
1




7






B


C B




C B




C








@
0
A @
1


A @


3
A








































i. Let v1; :::; vn 2 Rm and denote by A the matrix who's columns are
v1; :::; vn that is










1
0
j
j


j






A = @mvj1 vj2


vjn A
Denote: L(A) := fb 2 R
: (Ajb) has a solution g. Prove that L(A) =spanfv1; :::; vng.
ii. Let














B
1
2
1
C


A =
2
2
4
:
0
3
1
10
1


B






C




@
1
4
1
A













Find a basis for L(A) and the dimension of L(A).




Try to nd as 'fast' a solution as possible to each of these questions, us-ing dimension considerations, or any other theorems studied in class. In particular, when an actual computation is needed then use the technique of coordinates to solve the questions. Justify all your considerations, includ-ing the use of coordinates.



i. Is the following set a basis for
M2 3(R)?














1
2
(
1
1
2
) ;
1
1
2
) ; (
1
1
2
)g
f( 1
13 3);
1
3
3
( 1
3 3
1
3
3
ii. Is the following set linearly independent?
2 )
g










f(
0
1
) ; (


1
1
) ; (
2












1
2




1
1


1
1



















1
subspace of Rm.







2




iii. Is the following statement correct?
(
1
3
) 2 spanf(
0
1
) ; (


1
1


1
2








1
1
) ; (
2
2
)g
1
1


1
1








iv. Is the following set a basis for
M2(R)?
; (
2 2 )g




f( 1
3 )
;(0 1
) ;
( 1
1 )






1
1
1
2


1
1


1
1


v. Is the following statement true?








3x3g
2 + x2
2x3 2 spanf1
2x + x2 x3; 5 + 2x + 2x2
5x3; 3 + 6x
vi. Is the following set a spanning set for
R3[x]?






f2 + x2
2x3; 1 2x + x2


x3; 5 + 2x + 2x2
5x3; 3 + 6x 3x3g
vii. Find the dimension of the following space:






3x3g
spanf2 + x2
2x3; 1
2x + x2
x3; 5 + 2x + 2x2
5x3; 3 + 6x
viii. Is the following statement correct?
x3; 5+2x+2x2 5x3; 3+6x 3x3g
spanf2+x2
2x3; 3+6x
3x3g = spanf1
2x+x2



Find a basis for the following space:
(

f
a b + c
a + b + 4c d
a + 2b c 2d
a + b + c + 2d



)




: a; b; c; d 2 Rg



Let V be a vector space over R and let B be an ordered basis of V . Prove that if v 2 V and 2 R then [ v]B = [v]B.



(Remark: This was formulated as part of a theorem in class, but the proof was left for HW).




Let V be a vector space and B = (v1; :::; vm) be an ordered basis of V . Let W V be a subset of V and denote



[W ]B = f[w]B : w 2 W g;







so that [W ]B is a subset of Rm ([W ]B Rm). Prove that W is a subspace of V i [W ]B is a




(Remark: This was formulated as part of a theorem in class, but the proof was left for HW).




The following claims are either true or false. Determine which case is it for each claim and prove your answer.



Let V be a vector space which satis es dim V =3. Then there exist, a subspace W of V and a subspace U of W (that is, U W V ) such that dimU=1 and dimW =2.






3




Let V be a vector space which satis es dim V =3 and let W be a non trivial subspace of V and U be a non trivial subspace of W (that is, U W V ) then dimU=1 and dimW =2.



iii Let V be a vector space which satis es dim V =3 and let v1; v2; v3 2

be such that fv1; v2g are linearly independent, fv2; v3g are linearly independent, and fv3; v1g are linearly independent. Then fv1; v2; v3g is a basis for V .
iv. Let V be a vector space and v ; :::; v 2 V then: fv ; :::; v g is linearly

( 1 )n 1 n

independent i dim spanfv1; :::; vng = n.

Let V be a vector space and let V1; V2; V3 V be such that V1 + V2 = V1 + V3 and dimV2 =dimV3 then V2 = V3. (The sum of two subspaces was de ned in previous HW's).