Name + Uffi: D
‘tn’5
C.Wei
Section (circle one):
S.Bagley
Math 2250-4
Super Quiz 2
November 1, 2013
1 a) What is the
of a collection of vectors v
,
1
2’
v?
.L
(1 point)
,
,
1
lb) What does it mean for vectors v
2’
...
v to be linearly independent?
(1 point)
()
1 c) What is a basis for a vector space/subspace W?
(1 pornt)
2a) Find a basis for the solution space to homogeneous matrix equation A
the left below, and its reduced row echelon form is on the right.
2
6
0
0
0
3
-6 1
4
F
67Q
9 -8 -25 5
-3
1(i)
IO
=
,
where A is the matrix on
1 3 0 -3 010
001
U
1
O
2
0 0 0
0 1
ooj
116j0
(6 points)
F_\
0
:t
xl
(
1-3
2
,c
\\x
-
43b
-b
I I
LJ L
I
j Q
o
L°i
I
i
LJ
2b) What is the dimension of the solution space in part 2a?
(1 point)
3) Consider the differential equation below for a function x(t), which could arise from an unforced massspring configuration:
x’’(t) +2x’(t) + lOx(t)=0.
3a) Find tlgieral solutiontd this homogeneous linear differential equation. Hint: use the characteristic
polynomial method to ffit fmd a basis.
(6points)
o —o
(-)
2
(r--ñ r9Q
z
-‘
3b) ‘Which of the three damping phenomena is exhibited by solutions to this differential equation?
(1 point)
I’—
p(rJ .
3c) Now consider the inhomogeneous DE
x’
(1)
+ 2 x’ (1) + 10 x(t) -20.
=—2 is a particular solution. Use this fact and your previous work to write down the
general solution to the inhornogeneous DE.
(2 points)
2
Notice that
3d) Solve the initial value problem
x’’(t) +2x’(t) + lOx(1)-20
x(0) = 1
x’ (0) = 3.
(6 points)
+ (
c3
tt)
)L’(oz3
3
-3
) cz