Linear Algebra - Fall 2000
TEST #3
1.
Name:
Let
A   a1
a2
a3

 1 1 2 1
 1 0 1 2 


a4    2 4 6 0  , B  b1 b2


 3 3 0 1
 0 1 1 1 


1  e1 , e2 , e3 , e4 , a4 ;  2  e3 , b3 , b5
a)

Find a basis for:(4 pt.each)
ColA
NulA
RowA (6pt.)
b) Does the basis for
c)
ColB span 3 ? Justify your answer.(4pt.)
How do you verify that
d) Find
 a2  .(6pt.)
  1
1 is a basis for  5 ?(4pt.)
b3 b4
 1 1 2 3 0 
b5    2 1 6 0 1  ;
 1 2 0 1 1 
Find the transition matrix for the basis of
f)
Find the transition matrix for the standard basis of  to the basis of
3
g) Find a basis for the span of
h) If
2.
ColB to the standard basis for 3 . (3pt.)
e)
 x
  2
ColB .(4pt.)
b , b , b  .(4pt.)
1
2
3
4
  8 , find x .(5pt.)
10 
Suppose the system Ax  0 has 50 unknowns and its solution space is spanned by
independent vectors.
a) What is the nullity of A ?(3pt.)
b) What is the rank of A ?(3pt.)
c)
What is the rank of the row space of A ?(2pt.)
d) Can the matrix A have size
50 15? Why? (5pt.)
e)
Can the matrix A have size
15  50? Why? (5pt.)
f)
Find
n where rowA  n .(4pt.)
15 linearly
1 0 3


3. Let A  1 1 2 . Find the following:


 1 1 2 
a)
Find the characteristic polynomial of A . (10pt.)
b) Find the eigenvalues of A . (2pt.)
c)
Find the basis of the eigenspace corresponding to each eigenvalue from part b). You can use your
calculator to find these vectors but for one of your eigenspaces make sure that you verify that the
answer you get would be the same as the answer you would get without the calculator! (10pt.)
d) Diagonalize matrix A if possible. Give some justification.(4pt.)
4.
Let
2
7 
y    , u    . Write y as the sum of a vector in span u and a vector orthogonal to u .
6
1 
(If you have time please leave in fraction form.) (10pt.)
