Exam #2, Math 205A (Linear Algebra)
This take-home exam is due on Friday, November 15 by 5 PM. (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 reduced row echelon forms may be done on MATLAB or a calculator, but please show
all other work.
1. (16 points) Find the complete solution of the system
12 321
23 132
32−11 1 8
x1
x2
x3
x4
x5
=
3
10
25
.
2. (20 points) Find a basis for each of the four subspaces associated with the matrix
A=
1 1 2 1
1 2 1 2
1−47−4
What is the factored form of Athat displays these bases?
3. (14 points) The vectors
1
2
1
1
,
4
−1
5
−2
and
2
1
3
2
span a 3-dimensional subspace of R4. Find an
orthonormal basis for it.
4. (18 points) Let ~v1=
1
3
4
3
,~v2=
2
1
2
−1
, and ~v3=
1
3
0
5
, and let Sbe the subspace of R4spanned by
~v1and ~v2. Find the matrix Pthat pro jects vectors in R4onto S, and the matrix Rthat reflects through S.
Find also the projection ~p of ~v3onto Sand the reflection ~r of ~v3through S.
5. (16 points) Explain how you can tell that P=1
31
2 5 2 2 5
5 28 5 5 −3
2 5 2 2 5
2 5 2 2 5
5−3 5 5 28
is a projection matrix. Find
a basis for the subspace Tof R5that Pprojects onto, and a basis for T⊥(the orthogonal complement of T).
6. (10 points) We proved in class that Pis a projection matrix if and only if Psatisfies the two conditions
(i) PT=P(Pis symmetric) and (ii) P2=P.
The object of this problem is to reduce these two conditions to the single condition (iii) PTP=P.
Show that if Psatisfies (i) and (ii) then it satisfies (iii), and that if Psatisfies (iii) then it satisfies (i) and
(ii). This would prove that Pis a projection matrix if and only if it satisfies (iii).
7. (6 points) Suppose Mis a 2 ×2 matrix which satisfies M2=MT. Must Mbe a projection matrix? If
so, explain why. If not, what other possibilities are there for M?
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)