NAME
Id. No,
Final Examination, Math 304 Fall 2012, Wilkerson Section 508.
December 7, 2011 – Two Hours – Blocker 161
No notes, books, calculators, music players, earphones, etc. Show all work.
Problem
Score
1.(10)
2.(20)
3.(20)
4.(20)
5.(10)
6.(10)
7.(10)
8.(10)
9.(10)
10.(15)
Total(150)
1. 10 points
Let


−1 0 0




B = −3 2 0 .


0 0 1
Find the eigenvalues of B.
1
2. (20 points)
Let
 


1
1 1
 


 


 2
1 −1
 , ~b =   .
A=
 


 3
1 0 
 


4
1 4
a)(15 points) Write out the normal equations used in solving for the least squares solution to A~x = ~b.
Use this to find the least squares solution to Ax = b.
b) Find an orthonormal basis of the column space Col(A) .(5 points)
2
4. (20 points) A linear transformation L : V → V is represented by an matrix. Let VE = R3 be
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. Let AE = 2 1 3  represent the linear transformation L in the E-basis. Write


1 1 0
out the linear transformation L as AF in the F -basis.
3
 
1
 
 
√
√
 
 
5. (10 points) Let S ⊂ R3 have the orthonormal basis u1 = 1/ 3  1  and u2 = 1/ 2 0 .
 
 
−1
1
T
Find the projection of the vector (1, 2, 3) onto S.

1

6. (10
Let P be the plane in R3 which contains the point (2, 3, 5) and is perpendicular

 points)
1
 
 
to N =  5 .
 
11
a) Write an equation for P .
b) Is (0, 0, 0) on P ? Is (1, 1, 1) on P ?
4
7. (10 points) Let A be a symmetric 2 × 2 matrix with eigenvectors (1, 1)T for λ = 2 and
(1, −1)T for λ = 3. Using the theorem that AT = DT for suitable T and diagonal D, find T , D,
and A.
8. (15 points).


1 1 1




A = 1 1 1


! 1 1
has eigenvalues 3 and 0. For each eigenvalue λ calculate the subspace Eλ of v with Av = λv , i.e,
Null(A − λId).
a) For λ = 0.
b) For λ = 3.
5
9. 20 points 
1


The matrix A = 2

3
a) 10 points. Give a


1 2 0 1






has
reduced
row
echelon
form
U
=
0 0 1 3.
4 1 5



0 0 0 0
6 2 9
basis for the null space of A.
2 1 4

b) 5 points. Give a basis for ColSpace(A).
c) 5 points. Does the equation
 
2
 
 
A~x = 3
 
5
have a soluition ? If so, how many and list them?
6