NAME:
MA35100
Not the second MA35100 Test
Spring 2013
Monday 1st April 2013
50 minutes
Do not work on the front of this cover page
There are 8 questions. Answer all the questions.
The questions are not of equal value. The total number of marks is 100.
You
MA 35100 HOMEWORK ASSIGNMENT #12 SOLUTIONS
Problem 1. pg. 229; prob. 8
Consider a linear transformation L(x) = A x from Rn to Rm , with ker(L) = cfw_0. The pseudoinverse L+ of L is the transformation from Rm to Rn given by
L+ (y ) = the least-squares solu
MA 35100 HOMEWORK ASSIGNMENT #3 SOLUTIONS
Problem 1. pg. 65; prob. 4 Interpret the following linear transformation geometrically: T (x) = 1 1 -1 1 x.
Solution: Looking at Theorem 2.2.4 on page 63 of the text, this matrix is in the form a -b b a where a=1
Problem 1. (3 points) Which of the following transformations must be linear?
T : R13 R12 such that T ([ x1 x2 x3 ]) = [ x1 + 2x3 1 5x2 2x3 ].
S : P2 P4 such that S(q( x ) = ( x2 + 1)q(3x ) x3 q ( x 3) + 1.
ab
K : R22 R such that K (
) = adbc.
cd
L : C
APPENDIX: SAMPLE PROBLEMS FOR MIDTERM II MATH
351
(1) For what values of d is the vector (2, 1, d) in the span of (1, 2, 1) and (2, 5, 1).
(2) Let S = cfw_v1 , v2 , v3 , where v1 := (1, 2, 2), v2 := (3, 2, 1) v3 = (7, 6, 4).
Determine if S is lineraly ind
SAMPLE TEST PROBLEMS FOR MA351
TURKAY YOLCU
Here we give some past test problems with brief solutions and you are to work out the details.
Problem 1. Consider the following system
3x y +
pz
=
1
6x 2y + (2p + 1)z =
3
9x 3y + (4 p)z = 5 q
For which values o
MA 35100 LECTURE NOTES: WEDNESDAY, APRIL 14
Gram-Schmidt Process: m-Dimensional Space Let V be a subspace of Rn with basis B = (v1 , v2 , . . . , vm ). We may construct an orthonormal basis A = (u1 , u2 , . . . , um ) via the following steps: #1. Choose a
SAMPLE PROBLEMS FOR MIDTERM 2 MATH 351
(1) Is T1 : M22 M22 , T1 (A) = A + AT a linear transformation? Is
T2 : M22 M22 , T2 (A) = AAT a linear transformation?
(2) Find a basis and the dimension of the vector space of all the matrices which
10
commute with
FINAL EXAM
MA351
Electronic devices must be turned o. Documents are not allowed. For the Exercises part, all
answers must be carefully justied.
1
Multiple choices, 40 points
1. If A is a n n matrix, such that A2 = A, then 1 and 1 are the only eigenvalues
MA351-YOLCU Sketch Solutions of the Practice Problems for Midterm Exam
1
Problem 1. Consider
x+yz = 3
2x + y + z = 0
x + (a2 7)z = a
For which values of a does this system have
a) no solution?
b) innitely many solutions?
c) a unique solution?
Sketch Solut
MA 351 Practice Problems for Midterm Exam 1
Problem 1. Consider
x+yz
=3
2x + y + z = 0
x + ( a2 7) z
=a
For which values of a does this system have
(a) no solution?
(b) in nitely many solutions?
(c) a unique solution?
Problem 2. Consider the following sys
MA351-YOLCU
Section:
FALL 2011
FINAL EXAM
Name:
FOR THE PROBLEMS REQUIRING YOUR WORK, YOU MUST SHOW ALL YOUR WORK TO RECEIVE CREDIT
and WRITE NEATLY. In other words, any problem SAYING SHOW YOUR WORK without any scratch work won't
receive any credit. Be s