Math 110.201: Linear algebra (Spring 2008)
STUDY SHEET FOR THE FINAL EXAM
Scope of the exam
Chapters 1–8 (minus
§
7.6), plus the additional topics covered in lecture. The ﬁnal exam is a comprehensive
exam covering the content of the entire course.
Some additional practice problems
In addition to the problems from the midterm review sheets, try the following problems to get additional
handson practice; any diﬃculties that you may have with them will help pinpoint gaps in your understand
ing/knowledge. An answer key is not provided because understanding how to check the correctness of your
solution is in itself a very good gauge of your grasp of the material. If you are really stuck, ask a friend, or
visit your TA or the instructor during oﬃce hours.
Be sure that you know how to do all previous homework and test problems that you either couldn’t do,
or hadn’t done correctly. I also suggest going through some of the true/false problems at the chapter ends.
1.
If
A
is a 3
×
4 matrix such that
A
1
2
3
4
=
1

1
0
,
A
1
1
1
1
=
2
3

1
,
A
1
0

1

2
=
a
b
c
,
then what must the numbers
a
,
b
,
c
be? (Are the numbers even constrained by these equations?) How large
could the dimension of the kernel of
A
be?
2.
Find a 3
×
3 matrix
A
whose 2eigenspace is spanned by [1
,
1
,
1]
>
, [1
,
0
,
1]
>
and whose 3eigenspace is
spanned by [2
,
1
,
0]
>
.
3.
Compute the determinant of the following matrices:
A
=
0
0
0
1
0
0
2
0
0
3
0
0
1
2
3
4
,
±
0

A
I

I
²
,
±
A

A
I

I
²
.
4.
Let
~x
= [

1
,
1]
>
,
~
y
= [2
,
1]
>
. If possible, ﬁnd a 2
×
2 matrix
A
that satisﬁes the following equalities:
~x
·
A~x
= 6
,
~x
·
A~
y
= 0
,
~
y
·
A~x
=

6
,
~
y
·
A~
y
= 3
.
(Something to think about: Can you compute