Math310: Final Exam
Spring 2006
Problem 1.(50 pts) Find the eigenvalues and the corresponding eigenspaces for the
matrix. Decide whether the matrix is diagonalizable or not. (Explain!)
a)
240
120
004
, b)6 1
−1 4 .
Problem 2. Let A=
−200
−530
−521
.
It is given that Ahas eigenvalues λ1=−2, λ2= 3 and λ3= 1 with corresponding
eigenvectors u1=
1
1
1
, u2=
0
1
1
and u3=
0
0
1
.
a) (15 pts) Write down a factorization A=XDX −1, where Dis diagonal.
b) (10 pts) Find A5.
c) (10 pts) Find Bsuch that B3=A.
d) (10 pts) If the map L:R3→R3is given by L(x) = Ax for any x∈R3, show that L
is a linear transformation.
e) (10 pts) What is the matrix representation of Lwith respect to the basis {u1, u2, u3}?
f) (20 pts) What is the matrix representation of Lwith respect to the basis {v1, v2, v3},
where v1=
3
2
6
, v2=
0
1
1
, v3=
2
2
5
.
