Prof. A. Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
Geometric interpretation of some linear transformations
1. Rotation-dilations
To interpret the linear transformation
T (x) =
a b
x
b
a
a
a
r cos
in polar coordinates:
=
. Then the matrix
b
b
r sin
geome
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
SOLUTIONS to EXAM 2
1. 12 points
12345
A = 1 0 1 2 3
23058
1 0 0 2 7
2
rref A = 0 1 0 3
5
0 0 1 0 1
2
(a) Find a basis for the image of A.
Columns of A corresponding to the pivot columns of rref A:
1
2
3
4
5
6
7
8
Name:
NORTHEASTERN UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 1230
FINAL EXAM
Spring 2001
Instructions: Put your name in the blanks above. Put your nal answers to each question in the
designated spaces. Calculators are permitted. A single sh
1
2
3
4
5
6
7
8
Name:
NORTHEASTERN UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 1230
FINAL EXAM
Spring 2001
Instructions: Put your name in the blanks above. Put your nal answers to each question in the
designated spaces. Calculators are permitted. A single sh
Prof. A. Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
SAMPLE EXAM 2
2
1
10
4
2
2 0
1. Let A =
6 3 0 1.
2
1 2 0
(a) Find a basis for im A.
(b) Find a basis for ker A.
(c) Find rank A.
1 34
2. Let A = 4 5 2.
1 3 8
(a) Determine whether the column vectors of A
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
SOLUTIONS to EXAM 1
1. 10 points
(a) Use Gauss-Jordan elimination (rref) to solve the following system of equations. Carry out
the elimination all the way, before solving.
x1 + 2x2 + 3x3 + 4x4
2x1
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
EXAM 4
1. 12 pts
(a) Compute the area of the region enclosed by the folowing quadrilateral:
1
4
4
2
3
2
2
1
2
0
2
1
(b) Compute the area of the parallelogram spanned by the vectors and .
1
1
3
1
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
EXAM 1
1. 10 points
(a) Use Gauss-Jordan elimination (rref) to solve the following system of equations. Carry out
the elimination all the way, before solving.
x1 + 2x2 + 3x3 + 4x4 = 5
2x1 + 5x2 + 7
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
EXAM 2
1. 12 points
12
1 0
Let A =
23
(a) Find a basis for
3
1
0
the
45
2 3.
58
image of A.
(b) Find a basis for the kernel of A.
(c) Find the rank and the nullity of A.
Spring 2001
MTH 1230
Exam 2
2. 16 point
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
EXAM 3
1. 10 pts Consider the independent vectors
1
1
1
1
v1 = , v2 = ,
1
1
1
1
0
0
v3 = .
1
1
Find an orthonormal basis cfw_w1 , w2 , w3 for the subspace of R4 which has cfw_v1 , v2 , v3 as a b
Prof. A. Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
SAMPLE EXAM 4
1. Let A =
13
. What is the area of the parallelogram spanned by the column vectors of 5I2 A?
26
2. Compute the area of the triangle with vertices at (1, 2), (5, 7), (3, 8).
3. Compute the a