Exam #1, Math 205B (Linear Algebra)
This take-home exam is due at class time on Monday, October 6. (Sooner is fine.) You may consult the
textbook (or any other book) and any class notes and handouts, but please do not discuss any details of
this exam with anyone except me! Please sign the bottom of this sheet and turn it in with your exam.
You may ask me questions about the exam, but I reserve the right to give unsatisfying answers. Please show
all work (though you are encouraged to check your answers on MATLAB or a calculator).
1. (12 points) Find the angle between the vectors
1
1
1
0
1
and
1
1
1
2
1
.
2. (20 points) Solve the system
145
128
382
x1
x2
x3
=
3
−5
−45
by any method (by hand).
3. (24 points) (a) Find the LU factorization of A=
212
445
469
.
(b) Use your answer to (a) to solve A~x =
6
18
18
.
(c) Find L−1and U−1.
(d) Use your answers to (c) to compute A−1.
4. (12 points) If A=
1200
3400
0056
0078
, find A−1. If you guess the answer, show a check that it works, and
explain why you guessed what you did.
5. (20 points) Let R(φ) be the 2 ×2 matrix that reflects vectors in R2in the line making angle φwith the
positive x-axis. On the first project you looked at the product of two reflections. This time I want you to
try the product of three reflections, say
R(a)R(b)R(c) = µcos 2asin 2a
sin 2a−cos 2a¶µcos2bsin 2b
sin 2b−cos 2b¶µcos2csin 2c
sin 2c−cos 2c¶.
Is this another reflection? If so, in what line? If not, what is it? How many different orders can you multiply
these matrices together in, and how many of the products are genuinely different? (Hint: you should only
have to multiply them all together once. You can then get the other products just by switching a, b, c
around.)
6. (12 points) Let Ondenote the n×nzero matrix, in which every entry is zero.
(a) Suppose that Ais a 2 ×2 matrix such that A3=³0 0
0 0 ´=O2. Prove that A2must also equal O2.
(b) If Ais an n×nmatrix with n > 2, then it can happen that A3=Onbut A26=On. Find an example.
(Note: this is probably easier than part (a), so please try it even if you got stuck there.)
I affirm that I did not receive help from another person in doing this exam, nor did I give help
to another student in the class.
(signed)