Prof. Alexandru Suciu
MTH U371
LINEAR ALGEBRA
Spring 2006
QUIZ 4
301
1. 6 points Let A = 0 3 4.
1 5 7
Determine whether the column vectors of A are dependent or independent. If they are independent, say why. If they are dependent, exhibit a linear depende
Prof. Alexandru Suciu
MTH U371
LINEAR ALGEBRA
Spring 2005
Sample Questions for Quiz 3
2
1
4
2
1. Let A =
6 3
2
1
(a) Find a basis for
10
2 0
.
0 1
2 0
im A.
(b) Find a basis for ker A.
(c) Find rank A.
1234
1 0 1 2
2. Let A =
2305
(a) Find a basis for th
Prof. A. Suciu
MTH U371LINEAR ALGEBRA
Spring 2005
Sample Quiz 4, with Solutions
1. Let A be an orthogonal n n matrix (recall this means that the columns of A are orthonormal), and let A be its transpose.
(a) Find:
AA = In [This is equivalent to columns of
Prof. A. Suciu
MTH U371LINEAR ALGEBRA
Spring 2005
PRACTICE QUIZ 4
1. Consider the independent vectors
1
1
v1 = ,
1
1
1
1
v2 = ,
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 basis.
Prof. A. Suciu
MTH U371LINEAR ALGEBRA
Spring 2005
PRACTICE QUIZ 5
1 1 2 2
1 2
1
6
1. Use Gauss-Jordan elimination to nd the determinant of the matrix A =
2 6 10 33 .
2 2 5 10
2. Let A and B be two 5 5 matrices, with det A = 0 and det B = 3.
(a) Is A inve
Prof. A. Suciu
MTH U371LINEAR ALGEBRA
Spring 2005
PRACTICE FINAL EXAM
1. Are the following vectors independent or dependent? If they are independent, say why. If they are
dependent, exhibit a linear dependence relation among them.
3
v1 = 0
1
1
2. Conside
1
2
3
4
5
6
7
8
Name:
NORTHEASTERN UNIVERSITY
DEPARTMENT OF MATHEMATICS
FINAL EXAM
MTH U371
Spring 2005
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. Alexandru Suciu
LINEAR ALGEBRA
MTH U371
Spring 2005
QUIZ 2
1. 5 points Use Gaussian elimination to nd the inverse of following matrix. Indicate for each
step which row operation you use.
101
A = 2 4 0
1 1 1
2. 5 points For which choices of the cons
MTH U371
Prof. Alexandru Suciu
LINEAR ALGEBRA
Spring 2005
QUIZ 1
1. 7 points Use Gauss-Jordan elimination to solve the following system of equations. bf Indicate
for each step which row operation you use. Carry out the elimination all the way (to rref
for
Instructor: Prof. A. Suciu
MTH U371
Name:
LINEAR ALGEBRA
Spring 2005
QUIZ 4
1. 8 points Apply the Gram-Schmidt process to the vectors v1 =
the result in the form A = Q R.
3
2
, v2 =
, and write
1
0
MTH U371
Quiz 4
Spring 2005
2. 7 points
4
1
1
0
Consider
Instructor: Prof. A. Suciu
MTH U371
Name:
LINEAR ALGEBRA
Spring 2005
QUIZ 5
1. 8 points Let A, B , C be three 4 4 matrices, with det A = 2, det B = 1, det C = 0.
(a) In the following, circle the correct answer. Provide a (short) explanation in each case.
Prof. Alex Suciu
MTH U371
LINEAR ALGEBRA
Spring 2005
Solutions to Quiz 4
1. 8 points Apply the Gram-Schmidt process to the vectors v1 =
3
2
, v2 =
,
1
0
and write the result in the form A = Q R.
r11 = v1
=
9+1=
1
v1
r11
=
=
1
10
10
13
10 1
r12 = u1 v2
u1
Prof. A. Suciu
MTH U371 LINEAR ALGEBRA
Spring 2005
SOLUTIONS to QUIZ 3
1 02 0 3
1. 12 points
Let A = 2 0 4 1 7.
1 3 0 6 2
(a) Find the row reduced echelon form of A.
10 2 0
3
rref A = 0 1 2/3 0 11/3 .
0 0 0 1 1
(b) Find a basis for the image of A.
Columns
Prof. Alexandru Suciu
LINEAR ALGEBRA
MTH U371
Spring 2005
QUIZ 2
1. 5 points Use Gaussian elimination to nd the inverse of following matrix. Indicate for each
step which row operation you use.
101
A = 2 4 0
1 1 1
Compute:
.
.
rref
[A . I3 ] [I3 . A1 ]
.
Prof. Alexandru Suciu
MTH U371
LINEAR ALGEBRA
Spring 2006
QUIZ 1
1. 6 points Use Gauss-Jordan elimination to solve the following system of equations. Indicate
for each step which row operation you use. Carry out the elimination all the way (to
rref form)
Prof. Alexandru Suciu
MTH U371
LINEAR ALGEBRA
Spring 2006
QUIZ 6
1. 9 points
(a) Find the least squares solution x of the inconsistent system Ax = b, where
20
3
1 1 and b = 2
A=
01
1
(b) Use your answer to part (a) to nd the projection of b onto im A.
(c
Prof. Alexandru Suciu
MTH U371
LINEAR ALGEBRA
Spring 2006
QUIZ 2
1. 2 points Find the matrix A of the linear transformation T : R4 R3 given by
y1 = x1 + 3x2 + 2x3
y2 = 5x2 + 2x3 x4
y3 = 7x1 + x2 4x3 + x4
2. 4 points Consider the linear transformation T :
Prof. Alexandru Suciu
MTH U371
LINEAR ALGEBRA
Spring 2006
QUIZ 3
1. 3 points Sketch the image of the unit square under the linear transformation
T (x) =
1 2
x
13
2. 8 points Find the matrices of the following linear transformations:
(a) T : R2 R2 , a cloc
Prof. Alexandru Suciu
MTH U371
LINEAR ALGEBRA
Spring 2006
QUIZ 5
1. 8 points Apply the Gram-Schmidt process to the vectors v1 =
4
1
, v2 =
, and write
3
2
the result in the form A = Q R.
1
2
2. 6 points Consider the vectors v = 1 and w = 5 .
2
5
(a) Find
1
2
3
4
5
6
7
8
Name:
NORTHEASTERN UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH U371
FINAL EXAM
Spring 2006
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. Alexandru Suciu
MTH U371
LINEAR ALGEBRA
Spring 2006
QUIZ 7
4
1. 7 points Let A = 0
0
0
0
2
2 .
9 5
(a) Find the eigenvalues of A.
(b) Find a basis for each eigenspace of A.
(c) Find a diagonal matrix D and an invertible matrix S such that A = S D S
Prof. Alexandru Suciu
MTH U371
LINEAR ALGEBRA
Spring 2006
Solutions to Quiz 5
1. 8 points Apply the Gram-Schmidt process to the vectors v1 =
4
1
, v2 =
, and write
3
2
the result in the form A = Q R.
r11 = v1
=
16 + 9 = 5
1
v1
r11
=
14
53
r12 = u1 v2
=
14
Prof. Alexandru Suciu
LINEAR ALGEBRA
MTH U371
Spring 2005
QUIZ 1
1. 7 points Use Gauss-Jordan elimination to solve the following system of equations. bf
Indicate for each step which row operation you use. Carry out the elimination all the
way (to rref for
Instructor: Prof. A. Suciu
MTH U371
Name:
LINEAR ALGEBRA
QUIZ 3
1 02 0 3
1. 12 points
Let A = 2 0 4 1 7.
1 3 0 6 2
(a) Find the row reduced echelon form of A.
(b) Find a basis for the image of A.
(c) Find a basis for the kernel of A.
(d) Find the rank and