Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000
Quiz 3 –due Monday, September 20, 1999
Box
NAME
INSTRUCTIONS: You are to do your own work on this quiz - no help from anyone. You
may use your notes and my notes but do not discuss problems or solutions with anyone else.
There is no time limit for0taking this1quiz.
0
1
Ã
!
Ã
!
¡1 2
1 1 2
1 2
1 1 ¡3
C
C
Let A =
,B=B
,D=B
@ 1 ¡2 A, C =
@ 1 2 3 A
3 6
2 2 ¡6
2 ¡4
3 3 6
I. Give a basis and a description of the following spaces (eg a line in R3 )
1) The null space of A:
2) The column space of B:
3) The row space of C > :
4) The range of D:
0
1
0
C
!
5) The solution space for B ¡
x =B
@ 0 A
0
6) The null space of D> :
II. Answer the following questions (usingÃthe same
! matrices as above)
¡3
!
7) Describe the solution space of A¡
x =
:
¡9
!
8) Describe the solution space of C ¡
x =
Ã
0
1
1
!
:
1
0
C
!
9) How is the solution space of D¡
x =B
@ 1 A related to the null space of D?
0
10) Find bases for the null space of B and the row space of B: Show that these two spaces
are perpendicular to each other.
III. Use Gaussian elimination (by hand) to …nd the solution(s) to the following system of
equations. Do your work on scrap paper and then put your …nal (correct) steps here. BE
NEAT. You can use either equations or the augmented matrix. Put your answer in vector
parametric form
0
B
B
B
@
1
0
1 ¡3 2
2
1
B
B
2 ¡5 6
6 ¡1 C
CB
CB
1
0
8
9 ¡6 A B
B
@
¡2 7 ¡2 ¡1 ¡3
x
y
z
u
v
1
1
0
14
C
C
B 37 C
C
C
C=B
C
B
C
@ 40 A
C
A
¡20