Extra Pdf Problems
November 6, 2015
These problems are graded on an honor system. On Web Assign, there
is a problem worth 1 pt asking if you attempted a solution. Please make
sure to answer yes to receive the 1 pt.
1. Consider dierentiable n n matrix func
Math 308 H
Midterm 2
Print Your Name
Spring 2010
Student ID #
Problem
Total Points
1
8
2
6
3
12
4
8
5
6
6
6
Total
46
Score
Directions
Please check that your exam contains a total of 7 pages.
Write complete solutions or you may not receive credit.
This
Math 308 H
Midterm 1
Print Your Name
Spring 2010
Student ID #
Problem
Total Points
1
8
2
10
3
8
4
11
5
8
Total
45
Score
Directions
Please check that your exam contains a total of 6 pages.
Write complete solutions or you may not receive credit.
This exa
24. Markov Chains II. June 3, 2013
24.1. Formulation of the Problem
Assume there are only three airports in the US: Seattle-Tacoma (SEA), John F. Kennedy in New
York (JFK), and Los Angeles International Airport (LAX). Each day, 30% of planes in SEA y to
L
23. Markov Chains I. May 30, 2013
23.1. Formulation of the Problem
Assume there was only one fast-food restaurant in a small town: McDonalds. Every day customers
went to this restaurant. But then another fast-food place, a Subway, opened there. People sta
22. Quadratic Forms. May 28, 2013
22.1. Example
What kind of curve is
2x2 + 2x1 x2 + 2x2 = 1?
2
1
Let us invent some other Cartesian coordinates y1 , y2 , in which it will have really simple expression.
First, construct a matrix
2 1
A=
1 2
The rule is as
21. Diagonalization of a Matrix. May 24, 2013
21.1. Example
Let
A=
0 1
2 3
There are two eigenvalues, 1 = 1 and 2 = 2. Eigenvector corresponding to 1: v1 =
vector corresponding to 2: v2 =
1
Eigen1
1
.
2
x1
= y1 v1 + y2 v2 . Since v1 and v2 are linearly in
20. Defective Matrices. May 22, 2013
20.1. Starting Example
Find eigenvalues and eigenvectors for
0 1 0
A = 0 0 0
0 0 1
The characteristic polynomial is
det(A I3 ) =
1
0
0
0
= ()2 (1 ).
0
0 1
It has roots 0 and 1. Find eigenvectors corresponding to 0:
18. The Eigenvalue Problem. May 17, 2013
18.1. Starting Example
Try to calculate
A1 0x, A =
2 1
1
, x=
1 2
0
It is very hard to perform this directly. But suppose we had the vector v1 = [11]T instead of x; then
Av1 =
2 1
1 2
1
3
=
= 3v1 .
1
3
So it acts r
12. Review. April 26, 2013
12.1. Example
Consider the matrix
1 2 3 4
A = 2 3 4 5
3 4 5 6
(1) Let us nd its nullspace
N (A) = cfw_x R4 | Ax = 0.
Let us also nd a basis for this nullspace, and a dimension dim N (A), which is called the nullity for
the matri
15. Orthogonal 2 2-matrices. May 10, 2013
15.1. Rotations
Consider the matrix
A=
0 1
1 0
How does it act on vectors x R2 ? We have:
x=
x1
x2
Ax =
x2
x1
For example, if x = [1, 0]T , then Ax = [0, 1]T ; if x = [0, 1]T , then Ax = [1, 0]T ; if x = [1, 1]T
Math 308 B
Final Exam
Print Your Name
Autumn 2009
Student ID #
Problem
Total Points
1
18
2
11
3
7
4
8
5
14
6
6
7
6
8
10
Total
80
Score
Directions
Please check that your exam contains a total of 9 pages.
Write complete solutions or you may not receive cr
Math 308 Solutions
In
3
E=
2
1.6.1
Sol.
1.6.3
Sol.
1.6.7
Sol.
1.6.11
Sol.
1.6.13
Sol.
1.6.15
1.6.30
Sol.
1.6.31
Sol.
Sec. 1.6
Problems(1,3,7,11,13,15) page 1
3
each of Sec. 1.6, 1,3,7,11,13,15, use the appropriate matrices from the list: A = 4
2
1 1
1
3
Extra Pdf Problems
October 30, 2015
These problems are graded on an honor system. On Web Assign, there
is a problem worth 1 pt asking if you attempted a solution. Please make
sure to answer yes to receive the 1 pt.
1. Consider constants a, b satisfying a
Extra Pdf Problems
October 22, 2015
These problems are graded on an honor system. On Web Assign, there
is a problem worth 1 pt asking if you attempted a solution. Please make
sure to answer yes to receive the 1 pt.
1
2
1. Let h be a constant, and let v1
Extra Pdf Problems
October 9, 2015
These problems are graded on an honor system. On Web Assign, there
is a problem worth 1 pt asking if you attempted a solution. Please make
sure to answer yes to receive the 1 pt.
1. Recall from high school algebra that a
Extra Pdf Problems
September 30, 2015
These problems are graded on an honor system. On Web Assign, there
is a problem worth 1 pt asking if you attempted a solution. Please make
sure to answer yes to receive the 1 pt.
1. Recall from high school algebra tha
Math 308 Solutions
Sec. 2.2
Problems(1,7,9,11,18,19,28,32) page 1
In 1,7 determine if the given W , a subset of R2 , is a subspace. If it is, give a geometric description of W.
2.2.1
W = cfw_x: x 1 = 2x 2 . This is a subspace. Check the 3 properties:
0
=
Math 308 Solutions
Sec. 2.3
Problems(15,19,21(a,b,c),25,35,40) page 1
2
0
1
1
2, w = 1, x = 1, y = 2. You
15,19 refer to some sets S formed from some of the vectors: v =
2
1
1
0
are to either show that Spcfw_S = R3 or give an algebraic specica
Math 308 Solutions
Sec. 1.9
Problems(3,7,11,22,25,27,33,38,41) page 1
11
0 1 3
1 2
5 5 4 is the inverse of A = 1
1.9.3 Verify that the matrix B =
3 15 by showing that
5
1 1 1
0 1
AB = BA = I .
Solution
1 0 2 5 + 11 1 1 1 2 5 + 11 1 1 3 2 4 + 11 1
AB
Math 308 Solutions
1.6.21
Sol.
1.6.24
Sol.
1.6.27
Alt. Sol
1.6.33
Solution
1.6.35
Sec. 1.6 Problems(2l,24,27,33,35,41,48,50,57,62(b) page 1
3 1
Calculate the scalar |Au| where A = 4 7 and u =
2 6
3 - l + 1 - l 2
Au 2 4 - l + 7 - l = 3 . So taking the squ
Math 308 Solutions
1.8.6
Problems(6,8,12, 19, 27) page pN]
Sec. 1.8
Find the interpolating polynomial for the data in the table:
t
y
-2
-3
-1
1
1
3
2
13
Solution Since there are 4 data points (ts), we will need a 3rd degree polynomial to interpolate. So
y
17. Least Squares II. May 15, 2013
17.1. Least Distance to Subspace
Find the smallest distance from the vector
1
2
b=
3
to the plane W = cfw_x R3 | x1 + x2 + x3 = 0. Let us nd the basis for this subspace:
x2 x3
1
1
= x2 1 + x3 0
x = x2
x3
0
1
so t
16. Least Squares I. May 13, 2013
16.1. Linear Fit
Assume we conduct some experiment. We input the magnitude t and measure the output y, and we
have:
t
y
0
0
1
0
2
1
We would like to approximately nd y(t) in the form t + , where , are some coecients. We
c
14. Orthogonal Bases. May 6, 2013
14.1. Denition
A basis is called orthogonal if any two its vectors are perpendicular. A basis is called orthonormal
if, in addition, each vector has length one (so it is a unit vector).
1
1
1/2
2
and
For example, v1 =
and
Quiz 2 Solution
Math 308B. April 19, 2013
Consider the vectors
0
1 ,
v1 =
1
1
0 ,
v2 =
1
1
1
v3 =
0
(a) Are these vectors linearly dependent?
(b) Describe the set of vectors representible as linear combinations of v1 , v2 , v3 . This is called
the span of