MATH 223: Some results for 2 2 matrices.
One can preview a lot of the theory in this course by looking at the special case of 2 2
matrices. The proofs are relatively easy in this limited context and n
MATH 223: Some results for 2 2 matrices.
Multiplicative Inverses
It would be nice to have a multiplicative inverse. That is given a matrix A, nd the inverse
matrix A1 so that AA1 = A1 A = I. Such an i
MATH 223 Systems of Dierential Equations including example with Complex Eigenvalues
First consider the system of DEs which we motivated in class using water passing through two
tanks ushing out salt.
MATH 223
Complex Numbers
When solving a quadratic (over R) you may nd
there are no roots but you notice that you get
expressions involving 1. Rather than interpret 1 as a number, we can proceed as fol
Math 223
Symmetric and Hermitian Matrices.
An n n matrix Q is orthogonal if QT = Q1 . The columns of Q would form an orthonormal
basis for Rn . The rows would also form an orthonormal basis for Rn .
A
MATH 223: Extremal Set Theory
I would like to demonstrate one proof of a result of Sauer, Perles and Shelah, Vapnik and
Chervonenkis from 1971,1972. The proof is due to Smolensky and is from 1997. The
MATH 223. Vectors and Geometry.
We have already seen that geometry shows up strongly in linear algebra in the rotation matrix
R(). There are further remarkable interactions that are important in many
MATH 223. Orthogonal Vector Spaces.
Let U, V be vector spaces with U V . We consider
U = cfw_v Rn : for all u U, < u, v >= 0
Theorem 0.1 U is a vector space.
Proof: We have that U T is a vector space.
MATH 223
1. Let A =
Assignment #1 (2 pages)
Due Friday September 18, 2014 at start of class
x 1
0 1
and B =
. Determine all x, y so that AB = BA.
1 2
x y
2. Find a 2 2 matrix A with A2 = A where A = 0
MATH 223
Complex Numbers II
Let z = a + bi and w = c + di. We dened
zw = (a + bi)(c + di) = (ac bd) + (ad + bc)i
There are some interesting observations about this product. It is often the case that c
MATH 223: A Putnam Problem using our Linear Algebra.
(This problem came from a 1994 Putnam Problem.)
Problem: Let A,B, be two integer 2 2 matrices (i.e. 2 2 matrices with integer entries). Assume
they
MATH 223
Assignment #2
Due Friday September 25 at start of class
(two pages)
1.
a) Assume we have two nonzero vectors u, v. Let M = [u v], the 2 2 matrix with column 1
being u and column 2 being v. Sh
MATH 223
Assignment #3
due Friday September 26.
I like to see the solutions to a system of equations in Parametric Vector Form (or Vector Parametric
Form if you wish). If we nd a set of solutions:
x1
MATH 223
Assignment #4
due Wednesday October 7 in class.
Extra oce hours Tuesday 5-6. Midterm is scheduled for Friday October 9. A practice midterm is
posted.
1. Compute
3 6 2
i) det 4 7 ,
1 2 0
2
1
i
MATH 221 - FINAL EXAM
University of British Columbia
December 15, 2015.
First name:
Section number:
Last name:
Student number:
102 - MWF 10-11 am - Justin Tzou
103 - MWF 1-2 pm - Daniel Coombs
104 - M
Linear Algebra
MATH 223
2017W
MATH 223: Linear Algebra
Sven Bachmann
September December 2017
Basic information
Course website:
http:/www.math.ubc.ca/~sbach/teaching/2017W/223/Info.html
Time and room
MATH 223, 2017W
Homework set 4 due October 11th
Problem 1. (i) Let V1 , V2 , V3 be three real vector spaces and let : V1 V2 and : V2 V3 be
two linear transformations. Prove that their composition
: V
MATH 223, 2017W
Homework set 1 due September 20th
Problem 1. (i) Let V be a real vector space and let x V . Prove that its additive inverse is
unique.
(ii) Let now V be the vector space of real-valued