Final Exam, Math 205B (Linear Algebra)
This take-home exam is due by 12 noon on Friday, December 14. 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 appropriate place on the other side 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.
1. (14 points) Find the complete solution of the system
13111
11311
11131
x1
x2
x3
x4
x5
=
7
5
3
.
2. (26 points) Let A=
1 2 3
2 5 10
3 10 26
.
(a) Find the LU factorization of A. Is there anything special about it? Explain. (This might save you some
work later.)
(b) Find the determinant of Land the determinant of U.
(c) Use your answer to (a) to solve A~x =
9
31
83
.
(d) Use your answer to (b) to find the determinant of A.
(e) Find L−1and U−1.
(f) Use your answer to (e) to find A−1.
3. (19 points) Find a basis for each of the four subspaces associated with the matrix
A=
1 1 2 −2
1 2 1 5
2 1 1 −3
4 7 9 5
What is the factored form of Athat displays these bases?
4. (14 points) (a) Evaluate the determinant ¯
¯
¯
¯
¯
¯
¯
1 2 1 3
1 3 1 2
2 2 1 4
3 2 3 2
¯
¯
¯
¯
¯
¯
¯
.
(b) Are the vectors
1
1
2
3
,
2
3
2
2
,
1
1
1
3
and
3
2
4
2
linearly independent? Explain.
5. (14 points) Explain how you can tell that R=1
5
−1 2 2 4
2 1 −4 2
2−4 1 2
4 2 2 −1
is a reflection matrix. Find a
basis for the subspace Sof R4that Rreflects through, and a basis for S⊥.
6. (8 points) Suppose Pis a projection matrix onto some k-dimensional subspace Kof Rn. Describe the
eigenvalues and eigenvectors of Pas well as you can.
