MATH 348 - Advanced Engineering Mathematics January 26, 2009
Homework 3, Spring 2009 Due: February 2 , 2009
Linear Independence - Matrix Spaces - Vector Spaces
1. Determine the values of hfor which the vectors are linearly dependent.
v1=
1
−1
−3
,v2=
−5
7
8
,v3=
1
1
h
2. Given,
A=
−8−2−9
648
404
,w=
2
1
−2
.
(a) Is win the column space of A? That is, does w∈Col A?
(b) Is win the null space of A? That is, does w∈Nul A?
3. Given,
A=
2−3 6 2 5
−2 3 −3−3−4
4−6 9 5 9
−233−4 1
.
Determine:
(a) A basis and dimension of Nul A.
(b) A basis and dimension of Col A.
(c) A basis and dimension of Row A.
(d) What is the Rank of A?
4. Let,
v1=
1
0
−1
,v2=
2
1
3
,v3=
4
2
6
,w=
3
1
2
.
(a) How many vectors are in {v1,v2,v3}?
(b) How many vectors are in Span {v1,v2,v3}?
(c) Is wis in Span {v1,v2,v3}?
5. Given,
my00 +ky = 0, m, k ∈R.(1)
(a) Show that y1(t) = cos(ωt) and y2(t) = sin(ωt), where ω=qk
m, are solutions to the ODE.
(b) Show that any function in Span{y1, y2}is a solution to the ODE.1
1Since we know from differential equations that the only solutions to (1) are 0,y1, y2we can conclude that the space of solutions to (1) forms a
vector space whose basis is y1and y2.
1