NAME
Id. No,
Final Examination, Math 304 Fall 2011, Wilkerson Section.
December 13, 2011 – Two Hours – CE 222
No notes, books, calculators, music players, earphones, etc. Show all work.
Problem
Score
I.(30)
II.(30)
III.(20)
IV.(20)
V.(15)
VI.(20)
VII.(15)
Total(150)
1. 30 points
Let

2 0 0





B =  0 2 1 .


0 1 2
a) (10 pts.) One of eigenvalues of B is 1. Find the other two.
1
b) (10 pts) Find a nonzero eigenvector for each of the three eigenvalues.
c) (10 pts) Find an othgononal matrix Q and a diagonal matrix D so that D = QT BQ.
2
2. (30 points)
Let
 
1
1 1
 


 


2
 1 2
 , ~b =   .
A=
 


3
 1 3
 


5
1 4


a)(8 points) Write out the normal equations used in solving for the least squares solution to A~x = ~b.
b) (12 points) Find the least squares solution of A~x = ~b
c) Find an orthonormal basis of the column space R(A) and use this basis to find the projection
of ~b onto R(A).(10 points)
3
3.(20 points)
Let


−4 −3
 .
C=
10 7
a) Find the eigenvalues of C (10 points).
b) (10 points) Find two linearly independent eigenvectors of C.
4
4. (20 points) A linear transformation L : V → V is represented by a matrix after choosing a
basis of V . Let VE = R3 with the standard basis
  E = {e1, e2, e3 }.  
0
0
1
 
 
 






Let F be the basis of R3 consisting of v1 = 1 1 , v2 = 1 , v3 = 0 . Let VF denote R3 with
 
 
 
1
1
0


1 0 −1




the F -basis. Suppose that the matrix AE = 2 1 3  represents L in the E-basis. Write out


1 1 0
the linear transformation L as the matrix AF in the F -basis. That is, find AF
5
5. (15 points) Let


1 −1
 .
A=
1 1
a) (5 points) Find the eigenvalues of A.
b) (10 points) Find the singular values of A.
6
6. (20 points) Let


1 2 3 4




A = 2 1 7 4  .


5 7 16 16
a) Row reduce A. (6 points )
b) Give a basis for N(A). (5 points)
 
1
 


c) Solve A~x = ~b, for b = 5 for all possible ~x. (9 points)
 
8
7

1 2




3
7. (15 points) Let S ⊂ R be spanned by the column vectors of B = 5 4 . Find a basis for


9 3
the orthogonol complement of S.

8