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MATH 221H (Spring 2006) Final Exam, May 19th
No calculators, books or notes!
Show all work and give complete explanations for all your answers.
This is a 120 minute exam. It is worth a total of 100 points.
(1) [15 pts] Let A be the matrix


0 −1 −1
1 .
A = 1 2
1 1
2
(a) Show that the characteristic polynomial of A is p(λ) = −(λ − 1)2 (λ − 2).
/10 T
/100
(b) Calculate bases for each of the eigenspaces of A.
(c) Diagonalize the matrix A.
(2) [10 pts]
2
4
(a) Calculate the orthogonal projection of
onto
.
3
5
(b) Let B be the basis
 
1
1  
0 ,
b1 = √
2 1


1
1
b2 = √  1  ,
3 −1
 
1

Find the coordinate vector [x]B of x = 0.
0


2
1
b3 = √ −4 .
24 −2
(3) [15 pts] When Gaussian elimination is applied to

1
 2
[A|b] = 
 4
−1
we obtain the row echelon form
the augmented matrix

0 2
b1
1 0
b2 

1 4
b3 
2 −10 b4

0 2
b1

1 −4
b2 − 2b1
.
0 0 b4 − 2b2 + 5b1 
0 0 b3 − b2 − 2b1
 
1
2

(a) Is the system Ax = b consistent when b = 
 3 . Why?
−1

1
 0

 0
0
(b) Find bases for the column space and null space of A.
(4) [10 pts] Use row operations to calculate the determinant
1 a2 b + c 1 b2 a + c .
1 c2 a + b (5) [10 pts] Suppose that a 2×2 matrix A has eigenvalues λ1 = 12 and λ2 = 5 and corresponding eigenvectors
1
2
v1 =
,
v1 =
.
0
3
1
If xk = Axk−1 and x0 =
find an explicit formula for xk . Also calculate lim ||xxkk || and interpret your
4
k→∞
answer geometrically.
(6) [30 pts]
True or false? Explain why!
(a) If AB has 2 columns then A has two columns.
(b) If a 6 × 10 matrix is row equivalent to an echelon matrix with 4 non-zero rows, then the dimension of
the null space of A is 2.
(c) If A and B are square matrices with AB = BA and if A is invertible, then BA−1 = A−1 B.
(d) Let A be an n × n matrix. If there is a vector b ∈ Rn so that the equation Ax = b is inconsistent,
then the linear transformation T (x) = Ax is not one-to-one.
(e) Each eigenvector of a square matrix A is also an eigenvector of A2 .
(f) If the characteristic polynomial of a 3 × 3 matrix A is p(λ) = λ3 − 2λ2 + 7λ, then A has rank 3.
(7) [10 pts]
(a) Suppose that {v1 , v2 } is a linearly independent set in Rm . Show that {v1 , v1 + v2 } is also linearly
independent.
(b) Let u be vector in Rn . Show that the set of all vectors v that are orthogonal to u is a subspace of Rn .
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