Math 102
Midterm 1 - Oct 19
Name
Fall 2012
Student No.
Section A0
No aids allowed. Answer all questions on test paper. Total Marks: 20
[5]
1. Under what conditions on b (if any) does Ax = b have a solution?
1 2 3 5
A = 2 4 8 12
3 6 7 13
b1
b = b2
b3
Sol
Math 102
Quiz 1 - Oct 5
Fall 2012
Student No.
Name
No aids allowed. Answer all questions on test paper. Use backs of sheets if necessary.
Total Marks: 2
Given the matrix
2
1 1
A = 4 6 0
2
7 2
provide the elementary row operations that will put the matrix
Math 102
Quiz 5 - Nov 9
Student No.
Name
Fall 2012
Section A0
No aids allowed. Answer all questions on test paper. Total Marks: 2
Q1. Suppose that Cij are the cofactors of the 3 3 matrix A, that is, Cij = (1)i+j det(Mij ),
where Mij is obtained from A by
Math 102
Name
Quiz 3 - Oct 26
Student No.
Fall 2012
Section A0
No aids allowed. Answer all questions on test paper. Total Marks: 2
Q1. Let
V = spancfw_(1, 4, 4, 1), (2, 9, 8, 2)
nd a basis for V .
Solution: Knowing that N (A) = (C(AT ) , we know that our
Math 102
Name
Final Exam - Dec 14 - PCYNH 122 3-6pm
Student No.
Fall 2012
Section A0
No aids allowed. Answer all questions on test paper.
Total Marks: 40 8 questions (plus a 9th bonus question), 5 points per question.
The exam has 9 pages of questions, an
Math 102
Homework 1 - Oct 15
Fall 2012
1. If A = AT and B = B T , which of the following matrices are necessarily symmetric:
(a) A2 B 2
(b) (A + B)(A B)
(c) ABA
(d) ABAB
Solution: True for (a) and (c) only.
2. Let Tn be the set of n n upper-triangular mat
Math 102
Homework 4 - Nov 26
Fall 2012
1. Suppose the eigenvector matrix S has the property S T = S 1 . Show that A = SS 1
is symmetric and has orthogonal eigenvectors.
Solution: The orthogonality of the eigenvectors follows directly from the fact that
SS
Math 102
Homework 5 - Dec 7
Fall 2012
1. Diagonalize the following matrix:
A=
cos sin
sin
cos
Solution: Since det(A I) = (cos )2 sin2 it follows that
1 = cos + i sin
2 = cos i sin
Note that we may assume that 0 < < 2 since if = 0 the rotation by zero
Math 102
Midterm 2 - Nov 16
Student No.
Name
Fall 2012
Section A0
No aids allowed. Answer all questions on test paper. Total Marks: 15
[5]
1. Suppose that A = [ a b c ] where a, b, c are linearly independent column vectors.
Suppose that q1 , q2 , q3 are t
Math 102
Homework 2 - Oct 29
Fall 2012
1. Show that x y is orthogonal to x + y if and only if x = y .
Solution: Suppose that (x y) (x + y). Then,
0 = (x y)T (x + y) = xT x y T x + xT y y T y = xT x y T y,
and so x = y . The other direction follows directl