Exam #2, Math 205B (Linear Algebra)
This take-home exam is due on Friday, November 14 by 5 PM. 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 ļ£«
ļ£
13122
12143
14221
ļ£¶
ļ£ø
ļ£«
ļ£¬
ļ£¬
ļ£¬
ļ£
x1
x2
x3
x4
x5
ļ£¶
ļ£·
ļ£·
ļ£·
ļ£ø
=ļ£«
ļ£
3
4
2
ļ£¶
ļ£ø.
2. (20 points) Find a basis for each of the four subspaces associated with the matrix
A=ļ£«
ļ£¬
ļ£
1 1 3 5
2 1 2 7
2 3 10 13
5 3 7 19
ļ£¶
ļ£·
ļ£ø
What is the factored form of Athat displays these bases?
3. (14 points) The vectors ļ£«
ļ£¬
ļ£
1
1
1
2
ļ£¶
ļ£·
ļ£ø,ļ£«
ļ£¬
ļ£
1
2
1
5
ļ£¶
ļ£·
ļ£øand ļ£«
ļ£¬
ļ£
1
4
1
4
ļ£¶
ļ£·
ļ£øspan a 3-dimensional subspace of R4. Find an orthonormal
basis for it.
4. (18 points) Let ~v1=ļ£«
ļ£¬
ļ£
1
3
1
1
ļ£¶
ļ£·
ļ£ø,~v2=ļ£«
ļ£¬
ļ£
1
1
3
1
ļ£¶
ļ£·
ļ£ø, and ~v3=ļ£«
ļ£¬
ļ£
3
3
1
ā1
ļ£¶
ļ£·
ļ£ø, and let Sbe the subspace of R4spanned by
~v1and ~v2. Find the matrix Pthat projects 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. (18 points) Explain how you can tell that P=1
12
ļ£«
ļ£¬
ļ£¬
ļ£¬
ļ£¬
ļ£¬
ļ£
5 2 ā1ā1 2 5
2 2 2 2 2 2
ā1 2 5 5 2 ā1
ā1 2 5 5 2 ā1
2 2 2 2 2 2
5 2 ā1ā1 2 5
ļ£¶
ļ£·
ļ£·
ļ£·
ļ£·
ļ£·
ļ£ø
is a projection matrix.
Find a basis for the subspace Tof R6that Pprojects onto, and a basis for Tā„(the orthogonal complement
of T).
6. (8 points) Continuing a theme from the third homework set, construct a 4 Ć4 matrix whose column space
equals its nullspace.
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)
