# Exam # 1 Spring 2005 MATH 2270-01 Instructor: Oana Veliche ```Exam # 1
Spring 2005
MATH 2270-01
Instructor: Oana Veliche
Time: 50 minutes
NAME:
ID#:
INSTRUCTIONS
(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
Total
Max. # points
16
10
12
9
18
20
15
100
# Points
1
2
Problem 1. Consider the following matrix equation


1 0 1
 1 2 2 
0 1 1
A~x = ~b:
  
x
0
y  =  3 .
z
1
(8 points) (a) Find the solution to this system using elementary row operations.
(8 points) (b) Find the inverse of the matrix A.
3
(4 points) (c) Check that the matrix obtained in (b) is indeed the inverse of A.
(6 points) (d) For invertible matrices A, there is a unique solution to A~x = ~b, and a formula for this
solution which uses inverse matrix. Use this formula to resolve the system in part (a).
4
(6 points) Problem 2. Explain (in general) when an equation A~x = ~b, with A a n &times; m matrix, has a
unique solution?
(6 points) Problem 3. Suppose that T : R3
columns of the matrix A from Problem 1:

 
1
0
, T 
T  1  =
1
0
 
0
Find T  3  .
1
→ R2 is a linear map and that we know what T does to the

 
1
0
−1
1






2
2
=
.
=
, and T
2
2
1
1
5
Problem 4. We consider a linear transformation T (~x) = A~x for the following matrix A with its reduced
row-echelon form:




1 3 2 −1 −1
1 3 0 0 −1
0
3  and rref (A) =  0 0 1 0
1 
A= 2 6 5
0 0 0
1
2
0 0 0 1
2
(5 points) (a) Find a basis for the image of T .
(4 points) (b)Express the fifth column of A as a linear combination of the basis vectors you found in
part (a).
6
(8 points) (c) Find a basis for the kernel of T . Verify that your set of vectors is linearly independent.
(10 points) Problem 5. Prove that any linear transformation T : R2 → R2 , that is given by a rotation
has the form T (~x) = A~x, where
a −b
, a2 + b2 = 1.
A=
b
a
7
Problem 6.
(4 points) (a) Give the definition of linearly independent vectors.
(6 points) (b) Give the definition of a subspace of Rn , and give an example of a subspace.