Download Homework 4

MATH 110: LINEAR ALGEBRA
SPRING 2007/08
PROBLEM SET 4
If V is a vector space over F, we will write dimF (V ) for the dimension of V when we wish to
emphasize the field of scalars. For example, dimC (C3 ) = 3 and dimR (C3 ) = 6.
1. Let W1 , W2 be subspaces of V such that V = W1 ⊕ W2 . Let W be a subspace of V . Show that
if W1 ⊆ W or W2 ⊆ W , then
W = (W ∩ W1 ) ⊕ (W ∩ W2 ).
Is this still true if we omit the condition ‘W1 ⊆ W or W2 ⊆ W ’ ?
2. For the following vector spaces V , find the coordinate representation of the respective elements.
(a) V = P2 = {ax2 + bx + c | a, b, c ∈ R}. Find [p(x)]B where
p(x) = 2x2 − 5x + 6,
B = [1, x − 1, (x − 1)2 ].
(b) V = R2×2 . Find [A]B where
2 3
1 1
0 −1
1 −1
1 0
A=
, B=
,
,
,
.
4 −7
1 1
1 0
0 0
0 0
(c) V = R2 . Let θ ∈ R be fixed. Find [v]B where
x
cos θ
− sin θ
v=
, B=
,
.
y
sin θ
cos θ
3. Let W1 and W2 be the following subspaces of R4 .
     
   
1
4
5 
1
2 





     
   

2
−1
1
−1
−1
 ,   ,   , W2 = span   ,   .
W1 = span 
3  3   6 
 1   4 










6
6
12
1
5
(a)
(b)
(c)
(d)
(e)
Find a basis of W1 ∩ W2 .
Find a basis of W1 + W2 .
Extend the basis of W1 ∩ W2 in (a) to get a basis of W1 .
Extend the basis of W1 ∩ W2 in (a) to get a basis of W2 .
From the bases in (c) and (d), obtain a basis of W1 + W2 .
4. Let W1 , W2 , W3 be subspaces of a vector space V .
(a) Show that
dim(W1 + W2 ) + dim(W1 ∩ W2 ) = dim(W1 ) + dim(W2 ).
(b) Suppose dim(W1 + W2 ) = dim(W1 ∩ W2 ) + 1. Show that either W1 ⊆ W2 or W2 ⊆ W1 .
(c) Show that
dim(W1 ∩ W2 ∩ W3 ) ≥ dim(W1 ) + dim(W2 ) + dim(W3 ) − 2 dim(V ).
Date: March 6, 2008 (Version 1.0); due: March 13, 2008.
1
5. Let V be a vector space over R. We have seen in Homework 1 Problem 2 that W = V × V may
be made into a vector space over C with appropriate addition and scalar multiplication. W is
called the complexification of V . Show that
dimC (W ) = dimR (V ).
2