MATH 304.504-505
NAME
Final Review
SIGNATURE
1 May, 2016
Write your full name legibly on the front page of this exam booklet.
This exam consists of 11 problems.
The use of calculators is not permitted on this exam.
Points
Possible
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
11
10
Total
110
Credit
1 May, 2016
MATH 304
1.
Final Review
Page 2
Let {v1 , v2 , . . . , vn } be a set of linearly dependent column vectors.
a) The matrix
..
.
A=
v1
..
.

..
.
v2
..
.

..
. 
vn 

..
.
···
has a nontrivial nullspace. Explain why.
b) The matrix

···
· · ·

A=

···
v1T
v2T
..
.
vnT

···
· · ·



···
may have only a trivial nullspace. Provide an example.
1 May, 2016
MATH 304
2.
Final Review
Page 3
Let A be a square matrix.
a) Show that the matrices A and S −1 AS have the same determinant.
b) Show that the matrices A and S −1 AS have the same characteristic polynomial.
1 May, 2016
MATH 304
3.
Final Review
Page 4
Let {u1 , . . . , un } be an orthonormal basis for Rn . Assume that A is a matrix that all the elements in
this basis as eigenvectors. Show that A is a symmetric matrix. (Hint: Think about diagonalization).
1 May, 2016
MATH 304
4.
Final Review
Page 5
(10 pts.) Find an orthonormal basis of R3 consisting of eigenvectors of the symmetric matrix


1 0 −1
A =  0 1 1 .
−1 1 0
1 May, 2016
MATH 304
Final Review
Page 6
1 May, 2016
MATH 304
5.
Let
a) What are the eigenvalues of A?
b) Find corresponding eigenvectors.
Final Review

1
0

A=
0
0
1
2
0
0
1
1
3
0

1
1
.
1
4
Page 7
MATH 304
Final Review
Page 8
c) Find an invertible matrix P and a diagonal matrix D such that A = P DP −1 .
1 May, 2016
MATH 304
6.
 
1
0
1

Let u1 = √2 
1 and u2 =
0
R4 .
Final Review
Page 9
 
0
1
1
√  . Notice that {u1 , u2 } forms an orthonormal set of vectors in
2 0
1
a) (5 pts.) Given
 
1
2

x=
3 ,
4
find the orthogonal projection of x onto the subspace spanned by u1 and u2 .
 
4
3

b) (5 pts.) If the orthogonal projection of the vector y = 
2 onto the subspace W = Span{u1 , u2 }
7
 
3
5

is p = 
3, find a unit length vector in the orthogonal complement of W .
5
1 May, 2016
MATH 304
7.
Final Review
Page 10
(10 pts.) Let V = C 2 [0, 1] be the vector space of twice continuously differentiable functions on the
interval [0, 1]. Put
W = f ∈ C 2 [0, 1] : f (0) = f (1) and f 0 (0) = f 0 (1) ⊆ V.
Does W form a subspace of V ? Explain.
1 May, 2016
MATH 304
8.
Final Review
Page 11
(10 pts.) Let

3
A = 1
1

−2
3
3
and
 
2
b = 2
3
b closest to b such that Ax = b
b is consistent.
Find the vector b
1 May, 2016
MATH 304
9.
Final Review
Page 12
     
1 
2


 1

4 3 0
 ,   ,   ⊆ R4 .
Let W = Span 
  4 1


 3


2
1
2
a) (5 pts.) Write a 4 × 4 matrix A such that Im(A) = W .
b) (5 pts.) With A as in part a),
   
 
1
2
1
4 √ 3 0
   

b = 4
3 + 2 4 − 1 ∈ W.
2
1
2
Find a vector x ∈ R4 such that Ax = b.
1 May, 2016
MATH 304
10.
Final Review
Page 13
Find an orthonormal basis B = {u1 , u2 , u3 , u4 } of R4 such that u1 , u2 ∈ Row(A) and u3 , u4 ∈
Null(A) where A is the matrix


1 1 1 1
A = 1 1 2 2 .
0 0 1 1
1 May, 2016
MATH 304
11.
Final Review
Page 14
Write True, False or Not Decidable for the following statements.
Let T : V → W be a linear transformation between two vector spaces and dim(V ) = 100 and
dim(W ) = 50.
a) dim(Ker(T )) = 10.
b) Im(T ) = W
Let A be the matrix that represents T with respect to a choice of bases in V and in W .
a) The rows of A are linearly independent.
b) The columns of A are linearly independent.
c) The matrix A has 100 columns and 50 rows.
1 May, 2016
MATH 304
12.
Final Review
Page 15
Find three matrices L, D and U , such that A = LDU where


1 1 2
A =  1 3 4
−1 3 1
and L is an upper triangular matrix with diagonal entries equal to 1, U is an upper triangular matrix
with diagonal entries equal to 1 and D is a diagonal matrix.
1 May, 2016