Exam # 2
Spring 2005
MATH 2270-01
Instructor: Oana Veliche
Time: 50 minutes
NAME:
ID#:
INSTRUCTIONS
(1) Fill in your name and your student ID number.
(2) Justify all you answers. Correct answers with no justification will not be given any credit.
(3) No books, notes or calculators may be used.
Page #
2
3
4
5
6
7
8
9
Total
Max. # points
12
16
12
12
10
18
10
10
100
# Points
1
2
Problem 1.
Let P1 be the space of all polynomials of degree ≤ 1 and consider the following two bases:
U = (1, t)
and
B = (1, t + 2).
If T : P1 → P1 is the linear transformation given by T (f (t)) = f (2t + 1), find the following:
(6 points) (a) The matrix B of T with respect to the basis B.
(6 points) (b) The change of basis matrix S from the basis B to the basis U.
3
(5 points) (c) Using that the matrix of T with respect to the basis U is A =
1 1
, verify the formula
0 2
AS = SB.
(5 points) (d) If S ∗ is the change of basis matrix from U to B. What is the relationship of S ∗ with A
and B? Just write a formula, do not prove it and do not compute S ∗ .
(6 points) (e) Is T an isomorphism? Justify your answer.
4
Problem 2.
Let V be the subspace of R3 given by V = span(~v1 , ~v2 ) where
 
 
4
3
and
~v2 =  0 
~v1 =  0 
3
0
(6 points) (a) Find an orthonormal basis U of V .


3 4
(6 points) (b) Write a QR-factorization of the matrix A =  0 0  .
0 3
5
(5 points) (c) Find the matrix of the orthogonal projection on V with respect to the basis U.
(7 points) (d) Using that the matrix of dot products of the vectors ~v1 and ~v2 is
 25
4
−
 81
27
~v1 · ~v1 ~v1 · ~v2
9 12
and its inverse is 
=

12 25
~v2 · ~v1 ~v2 · ~v2
1
4
−
27
9
 
3
find the least-square solution ~x∗ of the system A~x = ~b, where ~b =  5 .
9




6
(10 points) Problem 3. Prove that if ~u1 , ~u2 , . . . , ~um are orthonormal vectors in Rn , then they are linearly
independent.
First write clearly, the definition of orthonormal vectors.
7
Problem 4.
(4 points) (a) Define the nullity of a linear transformation.
(5 points) (b) Give an example of a linear transformation T : P2 → P2 of non-zero nullity.
(4 points) (c) Define the angle between two vectors in Rn .
(5 points) (d) Give an example of two vectors in R3 such that the angle between them is acute.
8
Problem 5. True or false? Justify all your answers (prove when it is true, or give an example in case it
is not true).
(5 points) (a) If A is an n × n skew-symmetric matrix and orthogonal, then A2 = −In .
(5 points) (b) If A is an 5 × 3 matrix, then rank(A) + nullity(AT ) = 5.
9
(5 points) (c) The dimension of the linear space of lower-triangular 3 × 3 matrices is 3. (Display a basis
in either case, true or false).
(5 points) (d) If W and V are linear spaces such that W ⊂ V , W 6= V and V has infinite dimension,
then W has finite dimension.