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 inner products in spaces
of functions. No new ideas are
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 to come up with four examples. In each case, nd a real m
Math 312, Fall 2012
Jerry L. Kazdan
Problem Set 1
Due: In class Thursday, Sept. 13. Late papers will be accepted until 1:00 PM Friday.
These problems are intended to be straightforward with not much computation.
1. Let S :=
25
.
13
a) Find S 1 .
2
5
1
3
b
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 computation.
1. Solve all of the following equations. [
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 problems below, you should also know how to solve all of the
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 with the formula (A + B )2 =
+ B 2 ? Prove that if thi
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 calculators
but you may always use one 3 5 card with hand
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 solve the following problems
from the text. Most are simp
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 Thurs. Mar. 13[Late papers will be accepted until 1:00 on F
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 lists the estimated number of
genes and the estimated numb
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 1 .
2
5
1
3
b) For which constant(s) is the matrix
c)
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 in terms of the columns of A : det A =
det(A1 , A2 , . .
Math 312, Fall 2012
Jerry L. Kazdan
Problem Set 2
Due: In class Thursday, Sept. 20. Late papers will be accepted until 1:00 PM Friday.
Lots of problems. Most are really short.
In addition to the problems below, you should also know how to solve the follow
Math 312, Fall 2012 Problem Set 3
Jerry L. Kazdan
Due: In class Thursday, Sept. 27 Late papers will be accepted until 1:00 PM Friday. These problems are intended to be straightforward with not much computation. 1. [Bretscher, Sec. 2.4 #37]. If A is an inv
Math 312, Fall 2012
Due:
Jerry L. Kazdan
Problem Set 4
In class Thursday, Oct. 4 Late papers will be accepted until 1:00 PM Friday.
In addition to the problems below, you should also know how to solve the following problems
from the text. Most are simple
Math 312, Fall 2012
Jerry L. Kazdan
Problem Set 5
Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
In addition to the problems below, you should also know how to solve the following problems
from the text. Most are simple
Math 312, Fall 2012
Jerry L. Kazdan
Due: In class Thursday, Oct. 25 Late papers will be accepted until 1:00 PM Friday. Remark: We have completed Chapter 5, Sections 5.1, 5.2, 5.3, and 5.4 (except for the QR
Factorization cfw_ which we will skip). Since Fa
Math 312, Midterm 1
Aaron M. Silberstein
February 12, 2013
You have 50 minutes to complete this midterm.
1. Let A : R30 R40 , B : R40 R7 and C : R7 R57 be linear transformations.
(a) (10 points). What are the possible dimensions of the image of C B A?
(b)
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 injective, as dim ker cannot be zero. C B A
cannot be surjec
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 of dimension n. In particular, any particular element
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 vector can be written as some linear combination of
the
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 using the observation
01
that J := 1 0 has the property
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: Tuesday, April 23 [Late papers will be accepted until 1:00 o
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. Ideally, we would like to choose a and b so that
a + bt1
Math 312, Spring 2014
Jerry L. Kazdan
Problem Set 4 Solutions
Due: In class Thursday, Thurs. Feb. 13. Late papers will be accepted until 1:00 PM Friday.
Reminder: Exam 1 is on Tuesday, Feb. 18, 9:0010:20. No books or calculators
but you may always use one
Math 312, Spring 2014
Jerry L. Kazdan
Problem Set 5 Solutions
Due: In class Thursday, Thurs. Feb. 27. Late papers will be accepted until 1:00 PM Friday.
For the coming week, please read Chapter 5, Sections 5., 5.2, 5.3 [except for pages 221223
on the QR F
Math 312, Spring 2014
Jerry L. Kazdan
Problem Set 6
Due: In class Thurs, March 6. Late papers will be accepted until 1:00 PM Friday.
1.
[Bretscher, Sec. 5.1 #26]
the subspace S
Find the orthogonal projection PS
0 1
49
of ~ := @49A into
x
0 1
01
3
2
v
of R
ODE-Diagonalize: Examples
Example 1 Let
A :=
4 1
1 4
and x(t) =
dx
= Ax
dt
x1 (t)
.
x2 (t)
with
Solve
1
.
0
x(0) =
(1)
Solution: The key observation is that if A were a diagonal matrix, this would be simple. Thus
we begin by nding the eigenvalues ad eigen
Math 210
Jerry L. Kazdan
Vectors and an Application to Least Squares
This brief review of vectors assumes you have seen the basic properties of vectors
previously.
We can write a point in Rn as X = (x1 , . . . , xn ). This point is often called a vector.