Math 312, Spring 2013
Jerry L. Kazdan
Problem Set 9
Due: In class Thursday, Apr. 11. Late papers will be accepted until 1:00 PM Friday.
Lots of problems. Fortunately many are short.
1. This asks you t
Math 312, Fall 2012
Jerry L. Kazdan
Problem Set 8
Due: In class Thursday, Nov. 8 Late papers will be accepted until 1:00 PM Friday.
Some of this is on the material in Bretscher, Sec. 5.5, concerning i
Math 312, Spring 2013
Jerry L. Kazdan
Final Exam: Wednesday, May 1, 12:00-2:00 in DRL A-1. Closed book, no
calculators, but you may use one 3 5 card with notes on both sides.
Problem Set 10
Due: Tuesd
Math 312
Jerry L. Kazdan
Least Squares - Weighted
Say we have some data (t1 , y1 ), . . . , (tk , yk ) , where we might think of t as time, and seek a straight
line y = a + bt that best ts the data. I
Some Applications of Linear Algebra
1. Given n linear equations in n unknowns how can you tell
a) when a solution exists?
b) if that solution is unique?
2. Linear maps F (X ) = AX , where A is a matri
Math 312
Jerry L. Kazdan
The Eigenvalues and Eigenfunctions of Lu := u
2
Let Lu := u where u(x) is in the space C0 [a, b] of twice continuously dierentiable
real-valued functions on the interval a x b
Math 312
Jerry L. Kazdan
Multiple Integral: Change of Variable
Say we have a multiple integral
K :=
ZZ
1
R2
[1 + (x + 2y 1)2 + (3x + y + 2)2 ]2
dx dy
(1)
and would like to make the change of variable
Math 312
Jerry L. Kazdan
ODE-Coupled
01
is an orthogonal reection across
10
the line x1 = x2 . The eigenvectors V have the property that Av = v for
some constant . On geometric grounds, under this ree
Math 312, Fall 2012
Jerry L. Kazdan
Kernel of Lu := u + 4u
00
Let Lu := u + 4u . This dierential equation describes the motion of a mass on a spring.
These notes complete the proof in class concerning
Math 312
Jerry L. Kazdan
Orthogonal Projection
Let V be an inner product space (that is, a linear space with an inner product) and let
x1 , x2 , . . . , xk be non-zero orthogonal vectors and let S be
PROOF OF SCHURS THEOREM
DAVID H. WAGNER
In this note, I provide more detail for the proof of Schurs Theorem found in
Strangs Introduction to Linear Algebra [1].
Theorem 0.1. If A is a square real matr
Math 312, Spring 2013
Jerry L. Kazdan
Problem Set 8
Due: In class Thursday, Apr. 4 Late papers will be accepted until 1:00 PM Friday.
1. Complex numbers, z = x+iy , can be represented perfectly as 22
Math 312, Spring 2013
Jerry L. Kazdan
Problem Set 7
Due: In class Thursday, Mar. 28 Late papers will be accepted until 1:00 PM Friday.
1. [Bretscher (5th edition, Sec. 5.5 #39] The following table lis
Math 312, Spring 2013
Jerry L. Kazdan
Remark: We have almost completed Chapter 5, Sections 5.1, 5.2, 5.3, and 5.4 (except for
the QR Factorization which we will skip).
Problem Set 6
Due: In class Thur
Math 312, Midterm 1 Solutions
Aaron M. Silberstein
February 13, 2013
1. (a) (10 points). 0 dim im C B A 7.
(b) (10 points). By rank-nullity, 23 dim ker C B A 30.
(c) (10 points). C B A cannot be injec
Math 312, Midterm 2
Aaron M. Silberstein March 22, 2013
You have 50 minutes to complete this midterm. If n is a positive integer, let Pn := cfw_f (x) R[x] | deg f 50 be the vector space of polynomials
Math 312, Spring 2013
Jerry L. Kazdan
Linear Combination, Span, Linear Dependent
and Independent, .
Linear space V with vectors v1 , v2 , . . . , vk
Linear Combination
a1v1 + a2v2 + + anvk
Span
Every
Math 312, Spring 2013
Jerry L. Kazdan
Properties of Determinants
Let A be an n n matrix with columns A1 , A2 ,. . . , An . Below are the properties of
the determinant of A . We will often write them i
Math 312, Spring 2013
Jerry L. Kazdan
Rust Remover [Due: Never]
1. Solve the following system or show that no solution exists:
x + 2y
=
1
3x + 2 y + 4 z =
7
2x + y 2z = 1
2. Let S :=
25
.
13
a) Find S
Math 312, Spring 2013
Jerry L. Kazdan
Problem Set 1
Due: In class Thursday, Jan. 17. Late papers will be accepted until 1:00 PM Friday.
These problems are intended to be straightforward with not much
Math 312, Spring 2013
Jerry L. Kazdan
Problem Set 2
Due: In class Thursday, Jan. 24 Late papers will be accepted until 1:00 PM Friday.
Lots of problems. Most are really short.
In addition to the probl
Math 312, Spring 2013
Jerry L. Kazdan
Problem Set 3
Due: In class Thurs. Jan 31 [Late papers will be accepted until 1:00 on Friday ].
1. Let
2
A
A
and
+ 2AB
commute.
both be n n matrices. What's wrong
Math 312, Spring 2013
Jerry L. Kazdan
Problem Set 4
Due: In class Thurs. Feb. 7 [Late papers will be accepted until 1:00 on Friday ].
Reminder: Exam 1 is on Tuesday, Feb. 12, 9:0010:20. No books or ca
Math 312, Spring 2013
Jerry L. Kazdan
Problem Set 5
Due: In class Thursday, Feb. 21 Late papers will be accepted until 1:00 PM Friday.
In addition to the problems below, you should also know how to so
Math 312, Spring 2014
Jerry L. Kazdan
Homework 3 Solutions
1. Let A, B , and C be n n matrices with A and C invertible. Solve the equation
ABC = I A for B .
Solution: B = A1 (I A)C 1 . You can rewrite
Math 312, Spring 2014
Jerry L. Kazdan
Homework 2 Solutions
1. [Bretscher, Sec. 1.2 #44] The sketch represents a maze of one-way streets in a city. The
trac volume through certain blocks during
an hour
Math 312, Spring 2014
1. Let A =
Jerry L. Kazdan
Homework 1 Solutions
2 5
. Compute the inverse of A and of A2 .
1 3
Solution
A 1 =
3
1
5
2
. Squaring this we nd the inverse of A2 is
14
(A )1 = (A1 )2
Math 312
Homework 2
Due Mon 9/18
Remember, no credit will be given for answers without justification.
Textbook Problems:
Problems
Section
9(a), 16, 22
2.1
1, 6, 12
2.2
1, 3, 14, 18
2.3
1, 14, 15
2.4
N
Math 312
Homework 3
Due Fri 9/22
Textbook Problems:
Problems
Section
2, 6
2.5
5, 9
2.6
16, 17
2.7
Non-Textbook Problems:
I) Consider the matrix equation AX = B, where A and B are the matrices
1 0
1 1
Final Exam
Math 312, Fall 2012, Prof. Jauregui
You have 120 minutes.
No books, phones, calculators, talking, etc. You are allowed your own hand-written 3 5
inch note card, both sides, but no other mat