MidTest/MAS3105
Page 3 of 5
3. (18 pts.)
(a) Write the general solution of the system
of equations
x
1
6
x
2
+8
x
3
=5
x
1
x
2
8
x
3
=
5
in parametric form.
(b)
If
A =
1
6 8
6
8
,
Nul(A) is a subspace of
k
for which k ?
k=
(c)
Using your results from part (a), give an explicit
description of Nul(A).
(d)
Is the linear transformation T:
3
→
2
defined by
T(
x
)=A
x
, where A is as in part (b), onto?
Explain.
(e)
When T is the linear transformation of part (d), what is the
kernel, ker(T), of T ?
Use all the information you have at hand.
ker(T) =
(f)
What can you say about the column space of the matrix A of
part (b)?
Why?
Explain as completely as possible.
Col(A) =
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Page 4 of 5
4. (10 pts.)
(a) For which value(s) of s does the following
equation, have a unique solution ?
s1 
2
x
1
1
0
s2
1
x
2
=
2
00 3
s
x
3
3
(b)
When the equation in (a) has a unique
solution, obtain x
2
in terms of s when
x
is that unique solution.
(Hint:
Cramer’s
rule might be simple enough here.)
5. (10 pts.)
Suppose that A isa6x6 matrix with
det(A) = 10.
(a)
When B is obtained from A by replacing row five with
negative twentynine times row three plus row five, then
det(B) =
.
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, linear transformation, Det

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