NAME
Id. No,
Second Midterm Examination, Math 304 Fall 2012, Wilkerson Section.
November 29, 2012 – 75 minutes
No notes, books, calculators, music players, earphones, etc. Show all work.
Problem
Score
I.(20)
II.(15)
III.(20)
IV.(20)
V.(12)
VI.(13)
Total(100)
1. 20 points
(15 points) Let


0 2

A=
−1 3
a) Give the charactaristic polynomial of A and its eigenvalues. (8 points)
b) Give a (nonzero) eigenvector for each eigenvalue. (12 points).
1
2. 15 points
The matrix


3 2 −2




A = 1 3 −1


1 2 0
has eigenvalues 1, 2, and 3. Find a nonzero eigenvector for each eigenvalue.
2
3. 20 points
Let
 


1
1 2
 


 


A = 1 3 , ~b = 2 .
 


5
1 4
Let v1 and v2 be the columns of A, and P its column space.
a) Calculate the projection of v2 onto v1 , projv1 (v2 ). (7 points)
b) Use this to calculate an orthonormal basis of P . (7 points).
 
1
 
 
c) Use this basis to calculate projP (b). for b = 2.
 
5
3
4. 20 points
Let

 

1
1 0

 

 ~  

A = 0 1 , b = 2 .

 

5
1 1
Let P be the column space of A.
a) Calculate the least squares solution y to Ax = b. 10 points
b) Calculate the projection of b onto P , projP (b). 5 points
c) Calculate the distance from b to P . 5 points
4
5. Let L : R3 → R2 be the linear transformation with L(1, 1, 0)T = (1, 0)T , L(1, 0, 1)T = (1, 2)T ,
and L(1, 1, 1)T = (2, 1)T . Find a 2x3 matrix A such that Ax = Lx. Hint: find L(1, 0, 0)T , etc. 12
points
6. (13 points) Let u = (1, 0, 1, 0)T and v = (1, 1, 1, 1)T . Let S be Span(u, v). Give a basis for the
orthogonal complement of S, S ⊥ .
5