Page 173 - Matrix Analysis & Applied Linear Algebra

168 Chapter 4 Vector Spaces

4.1.4. Why must a real or complex nonzero vector space contain an infinite
number of vectors?

3
4.1.5. Sketch a picture in of the subspace spanned by each of the following.
           
1 2 −3 −4 0 1
   
6
3
(a)      −9  , (b)  0      ,
,
,
,
5
,
1
2 4 −6 0 0 0
   
     
1 1 1
 
,
(c)       .
1
1
,
0
0 0 1
 
3
4.1.6. Which of the following are spanning sets for ?
(a) {(111 )} (b) {(100 ) , (001 )},
(c) {(100 ) , (010 ) , (001 ) , (111 )},
(d) {(121 ) , (20 −1) , (441 )},
(e) {(121 ) , (20 −1) , (440 )}.
4.1.7. For a vector space V, and for M, N⊆ V, explain why
span (M∪N)= span (M)+ span (N) .
4.1.8. Let X and Y be two subspaces of a vector space V.
(a) Prove that the intersection X∩ Y is also a subspace of V.
(b) Show that the union X∪ Y need not be a subspace of V.
m×n n×1
4.1.9. For A ∈ and S⊆ , the set A(S)= {Ax | x ∈S} contains
all possible products of A with vectors from S. We refer to A(S)as
the set of images of S under A.
n
m
(a) If S is a subspace of , prove A(S) is a subspace of .
(b) If s 1 , s 2 ,..., s k spans S, show As 1 , As 2 ,..., As k spans A(S).
4.1.10. With the usual addition and multiplication, determine whether or not
the following sets are vector spaces over the real numbers.
(a) , (b) C, (c) The rational numbers.
4.1.11. Let M = {m 1 , m 2 ,..., m r } and N = {m 1 , m 2 ,..., m r , v} be two sets
of vectors from the same vector space. Prove that span (M)= span (N)
if and only if v ∈ span (M) .

4.1.12. For a set of vectors S = {v 1 , v 2 ,..., v n } , prove that span (S) is the

intersection of all subspaces that contain S. Hint: For M = V,
S⊆V
prove that span (S) ⊆M and M⊆ span (S) .