NAME
Id. No,
Second Midterm Examination, Math 304 Fall 2011, Wilkerson Section.
November 14, 2011 – 50 minutes
No notes, books, calculators, music players, earphones, etc. Show all work.
Problem
Score
I.(20)
II.(20)
III.(20)
IV.(20)
V.(10)
VI.(10)
Total(100)
1. 20 points
(15 points) Let


1 2 1 7 1




A = 2 4 1 10 0


1 2 0 3 1

1 2 0


with rref form U = 0 0 1

0 0 0
a) Give a basis for the column
3 0



4 0.

0 1
space R(A).
b) Give a basis of the null space N(A).
1
c) Give a basis of the row space of A.
d) Give a basis of the orthogonal complement of the row space of A.
2. 20 points
Let
 


1
1 2
 


 


A = 1 3 , ~b = 2 .
 


5
1 4
a)(5 points) Write out the normal equations used in solving for the least squares solution to A~x = ~b.
b) (10 points) Find the least squares solution of A~x = ~b
c) Find the projection of ~b onto the column space of A.
2
3.(20 points)
a) Let W ⊂ R4 be spanned by
 
 
 
2
2
1
 
 
 
 
 
 
3
2
2





v1 =   , v2 =   , v3 = 
 
3
3
3
 
 
 
4
4
4
. Find a basis for W .
b) Suppose V is the subspace of R4 with basis
 
 
1
1
 
 
 
 
0
1
 , v2 =   .
v1 = 
 
 
1
1
 
 
1
1
Using the Gram-Schmidt procedure, convert these into an orthonormal basis of V .
3
4. (20 points) Let

  
3x3 − x2
x1

  

  
L x2  =  5x1 − x3  .

  
x1 + x2 + x3
x3
a) Find a 3x3 matrix A such that Lx~E = Ax~E , where x~E is ~x with respect to the standard basis
E = {e1 , e2 , e3 }.
 
 
 
0
0
1
 
 
 






b) Let F be the basis of R3 consisting of v1 = 1 1 , v2 = 1 , v3 = 0 . Find a 3x3
 
 
 
1
1
1
matrix B such that Lx~F = B x~F , where x~F is ~x written using the F -basis. Explain the thoery even
if you can’t do the calculation.
4
 
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 ?
5