Math 3333 Linear Algebra I Fall 2008
Name:
Homework 6 – Span and Linear Independence – due Tuesday, October 14st
You must show all of your work!
1 .) (2 points) (True or False) (circle one, no justification required ) If Wis a subspace of Vand
v1,v2,...,vnare vectors in W, then span(v1,v2,...,vn) is a subspace of W.
2 .) (4 points) Find a spanning set for the subspace
W={v∈R4such that Av=0}
where:
A=
121−1
2 1 1 1
2 1 1 1
3 .) (4 points) Find a spanning set for the subspace
W={p(t)∈P3such that Z1
0
p(t)dt = 0 and p(0) = 0}
4 .) (4 points) Are the following vectors in R3linear independent? If so explain why, if not
explicitly show why they are linearly dependent (i.e. write 0as a nontrivial linear combination of
these vectors).
v1=
1
1
1
,v2=
1
2
3
,v3=
0
0
1
,v4=
2
1
1
,
5 .) (8 points) Consider the following set of matrices in M2,2:
A1=1 1
0 2 , A2=0 1
1 0 , A3=2 0
0 1 , A4=1 0
1 1
a.) (4 points) Does {A1, A2, A3, A4}span M2,2? If so, show how to write any matrix a b
c d ∈M2,2
as a linear combinaton of {A1, A2, A3, A4}.
b.) (4 points) Is the set {A1, A2, A3, A4}linearly independent? Why or why not?
1