Prelim 1
Math 294
Spring 2004
no calculators
answer + reason = credit
condensed printout—there were 5 pages
1.
a)
(17 points)
Find the number
a
for which the system
x
1
+ 3
x
3

x
4
= 20
x
1
+ 2
x
2
+ 2
x
4
=
a
x
1
+ 4
x
2

3
x
3
+ 5
x
4
= 60
has solutions, and find all solution vectors
x
.
b)
(3 points)
Give a definition: “
u
1
,
u
2
, . . . ,
u
n
are linearly independent iff —”.
2.
(20 points)
Find the inverse
C

1
if
C
=
1
0
0
0
2
1
0
0
0
3
1
0
0
0
4
1
.
3.
a)
(8 points)
Find a basis for the kernel of
D
, if
D
is the 5 by 5 matrix having a 1 in every position.
b)
(8 points)
In this part, let
v
1
=

1

1
1
1
,
v
2
=

1
1
1

1
,
e
1
=
1
0
0
0
,
e
2
=
0
1
0
0
,
e
3
=
0
0
1
0
, and
e
4
=
0
0
0
1
. Find a basis for
R
4
consisting of
v
1
and
v
2
together with some of the
e
k
.
c)
(4 points)
Give at least 4 examples of vectors in the image, im
A
, where
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 Spring '05
 HUI
 Math, Linear Algebra, Algebra, Identity matrix

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