Exam #2, Math 205A (Linear Algebra)
This take-home exam is due at class time on Friday, March 22. (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. Matrix
multiplications and reductions to reduced row echelon form may be done on MATLAB or a calculator, but
please show all other work.
1. (16 points) Find the complete solution of the system
1312
2351
3418
x1
x2
x3
x4
=
2
4
20
.
2. (22 points) Find a basis for each of the four subspaces associated with the matrix
A=
11323
23734
12411
What is the factored form of Athat displays these bases?
3. (14 points) The vectors
3
2
1
2
,
1
5
−1
1
and
2
3
4
1
span a 3-dimensional subspace of R4. Find an
orthonormal basis for it.
4. (18 points) Let ~v1=
1
3
2
1
1
,~v2=
1
1
2
3
1
, and ~v3=
1
−1
4
3
1
, and let Sbe the subspace of R5spanned by
~v1and ~v2. Find the matrix Pthat pro jects vectors in R5onto S, and the matrix Rthat reflects through S.
Find also the projection of ~v3onto Sand the reflection ~r of ~v3through S.
5. (16 points) Explain how you can tell that P=1
20
1 3 3 1
3 11 9 −3
3 9 9 3
1−3 3 19
is a projection matrix. Find a
basis for the subspace Tof R4that Pprojects onto, and a basis for T⊥(the orthogonal complement of T).
6. (8 points) Find the line which best fits the four data points (1,2), (2,1), (3,3) and (4,2) in the sense of
least squares.
7. (6 points) Suppose Uis a 6-dimensional subspace of R8, and let Abe an 8 ×6 matrix whose columns
are a basis of U. To find the projection matrix onto Udirectly from the formula A¡ATA¢−1ATone would
have to invert a 6 ×6 matrix. Can you think of an alternative way to find this projection matrix where you
would only need to invert a 2 ×2 matrix?
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)