Math 2135 - Linear Algebra
Homework # Due Mar 20
1. Find the determinant of the following matrices.
(a)
1 0 2
0 2 0
1 0 3
(b)
1 1 2
1 0 1
1 1 0
2. What is the determinant of each of the following 2 2 matrices?
(a)
3 1
2 1
(b) Any rotation matrix.
(c)
MATH 2135, LINEAR ALGEBRA, Winter 2013
Handout 1: Lecture Notes on Fields
Monday, January 7, 2013
Peter Selinger
and multiplication is also called the logical and operation. For example, we
can calculate like this:
1 (1 + 0) + 1) + 1 =
=
=
=
1 (1 + 1) + 1
MATH 2135, LINEAR ALGEBRA, Winter 2013
Handout 2: What is a proof?
This handout summarizes some basic techniques used in everyday proofs. We
use the usual logical notations P Q for P implies Q, x A.P (x) for for
all x A, P (x), x A.P (x) for there exists
Math 2135 - Linear Algebra
Practise Mid-Term
1. For each of the following subsets W of a vector space V , determine if W is a subspace of V .
Say why or why not in each case:
(a) V = R3 and W = cfw_(a1 , a2 , a3 )|a1 3a2 + 4a3 = 0 , and a1 = a2
(b) V = R
Math 2135 - Linear Algebra
Homework #1 Due January 22
1. Find the inverse of the matrix
0 1 1
M = 1 1 0
0 0 1
using Z2 as the set of scalars. Hint: follow the usual steps of Gaussian elimination, but use
modulo 2 operatios. Compare this to the inverse of
Math 2135 - Linear Algebra
Homework #5 Due April 3
1. A linear transformation T : V V is called skew-adjoint or skew-Hermitian if T = T
(Recall T = T T the conjugate transpose). Prove
(a) The eigenvalues of a skew-adjoint mapping are purely imaginary.
(b
Math 2135 - Linear Algebra
Practise Final
Note: In addition the exam will have a short answer section for the rst question.
1. For each of the following subsets W of a vector space V , determine if W is a subspace. Justify
your answer.
(a) V = R3 and W =
Math 2135 - Linear Algebra
Homework #3 Due Feb. 20th
1. Show that equality holds in the Cauchy-Schwarz inequality if and only if u and v are linearly
dependent.
2. Let c1 , c2 and c3 be positive real numbers and let u = (u1 , u2 , u3 ) and v = (v1 , v2 ,
Math 2135 - Linear Algebra
Homework #2 Due Feb 6
1. Find a basis for and the dimension of the subspaces dened by the following conditions:
(a) (x1 , x2 , x3 , x4 ) R4 such that
x1 + x4 = 0 ,
3x1 + x2 + x4 = 0 .
(b) cfw_f Spancfw_ex , e2x , e3x | f (0) = f