In Linear Algebra II (Math 2275) picks up from where you left off in Linear Algebra I (Math 2255). We will learn about: eigenvalues and eigenvectors, orthogonality and least squares, symmetric matrices and quadratic forms, and if time, the geometry of vector spaces. We will also try to incorporate some software into the class.
Below is a summary of what we did in class, plus any relevant
news and/or information.
There are also some simple online calculators that allow you to do basic calculations. See
To download Octave, go to the
Once you have installed Octave, you should try some of the online tutorials:
Finally, here is another, more comphrensive tutorial:
####################################################################### ## Jan 26 Examples ## Math 2275 ###################################################################### ## Example from class ## attractor clear A = [.8 0; 0 0.64] b = [1; 1] # initial vector P1 = [b] for i=1:25 b = A*b; P1 = [P1, b]; end; c = [-1;1] # second initial vector P2 = [c]; for i=1:25 c = A*c; P2 = [P2, c]; end; d = [2;-3] # thider initial vector P3 = [d]; for i=1:25 d = A*d; P3 = [P3, d]; end; x1 = P1(1,:); y1 = P1(2,:); x2 = P2(1,:); y2 = P2(2,:); x3 = P3(1,:); y3 = P3(2,:); plot(x1,y1,"*",x2,y2,"*",x3,y3,"*") ## make a loop to do lots of points at once P = []; for i=1:30 # pick a random vecotr b = [100*(-1)^(ceil(10*rand()))*rand();100*(-1)^(ceil(10*rand()))*rand()]; P = [P b]; # for each vector, find trajector for i=1:30 b = A*b; P = [P, b]; end; # 4 iterations only end; x = P(1,:); y = P(2,:); # plot all trajectories plot(x,y,"*") ######################################################################## ## Example of repellor ## ## Example from class clear A = [1.2 0; 0 1.4] b = [1; 1] # initial vector P1 = [b] for i=1:25 b = A*b; P1 = [P1, b]; end; c = [-1;1] # second initial vector P2 = [c]; for i=1:25 c = A*c; P2 = [P2, c]; end; d = [2;-3] # third initial vector P3 = [d]; for i=1:25 d = A*d; P3 = [P3, d]; end; e = [1; -.01] # initial vector P4 = [e] for i=1:25 e = A*e; P4 = [P4, e]; end; x1 = P1(1,:); y1 = P1(2,:); x2 = P2(1,:); y2 = P2(2,:); x3 = P3(1,:); y3 = P3(2,:); x4 = P4(1,:); y4 = P4(2,:); plot(x1,y1,"*",x2,y2,"*",x3,y3,"*",x4,y4,"*") P = []; for i=1:30 b = [100*(-1)^(ceil(10*rand()))*rand();100*(-1)^(ceil(10*rand()))*rand()]; # pick random vector P = [P b]; for i=1:30 b = A*b; P = [P, b]; end; # 4 iterations only end; x = P(1,:); y = P(2,:); plot(x,y,"*") ########################################################################### ### Example of Sadle Point ### Exercise 9 from Section 5.6 clear A = [1.7 -.3; -1.2 .8] eig(A) P = []; for i=1:400 b = [100*(-1)^(ceil(10*rand()))*rand();100*(-1)^(ceil(10*rand()))*rand()]; # pick random vector P = [P b]; for i=1:3 b = A*b; P = [P, b]; end; # 3 iterations only end; x = P(1,:); y = P(2,:); plot(x,y,"*") |
# Basic Octave input
2+2
3-4
139/3
14*5.36
cos(pi*4)
cos(4)
# Inputting a matrix and solving a sytem of linear equations
# Example (Example 3, page 8)
m = [0 1 -4; 2 -3 2; 5 -8 7]
b = [8; 1; 1]
rref(m)
m\b
m*ans
# Example (Example 4, page 18)
A = [1 6 2 -5 -2; 0 0 2 -8 -1; 0 0 0 0 1]
b = [-4; 3; 7]
A\b
A*ans
# Example (Example 6, page 97)
A = [2 -5 0; -1 3 -4; 6 -8 -7; -3 0 9]
B = [4 -6; 7 1; 3 2]
A*B
transpose(A)
# Example (Example 7, page 108)
A = [0 1 2; 1 0 3; 4 -3 8]
inv(A)
A * ans
# Example (Exampe 2, page 126)
A = [2 4 -1 5 -2; -4 -5 3 -8 1; 2 -5 -4 1 8;-6 0 7 -3 1]
[l,u,p] = lu(A)
# gives you something different!
# Example (Example 3, page 155)
A = [2 5 -3 -4 8; 4 7 -4 -3 9; 6 9 -5 2 4; 0 -9 6 5 -6]
rank(A)
# Example (Example 3, page 166)
A = [3 -7 -8 9 -6;0 2 -5 6 3; 0 0 1 5 0; 0 0 2 4 -1; 0 0 0 -2 0]
det(A)
# Example (Example 6, page 285)
A = [5 0 0 0; 0 5 0 0; 1 4 -3 0 ; -1 -2 0 -3]
eig(A)
[a,b]=eig(A)
# Some plotting
# Example (Example 1, pg 295)
A = [0 -1;1 0] # Matrix
b = [2; 0] # Intial Vector
P = [b]
for i=1:20 b = A*b; P = [P, b]; end; # compute Ax_{i} = x_{i+1} for some values
x = P(1,:) # x coordinates
y = P(2,:) # y coordinates
plot(x,y) # plot x vs y
A = [0 -1;1 0]
b = [3;4]
P = [b]
for i=1:20 b = A*b; P = [P, b]; end;
x = P(1,:)
y = P(2,:)
plot(x,y)
# Example (Example 3, pg 297)
A = [.5 -.6;.75 1.1]
b = [2; 0]
P = [b]
for i=1:20 b = A*b; P = [P, b]; end;
x = P(1,:);
y = P(2,:);
plot(x,y,"*")
# Example
clear
A = [.8 .5; -.1 1]
b = [.13; -.23]
P = [b]
for i=1:100 b = A*b; P = [P, b]; end;
x = P(1,:);
y = P(2,:);
plot(x,y,"*")
# Example from class
clear
A = [1 -2; 1 3;]
[a,b]=eig(A)
b = [1; 1]
P = [b]
for i=1:200 b = [1/sqrt(5) 0; 0 1/sqrt(5)]*A*b; P = [P, b]; end; ### I added a correcting factor!
### the matrix A streches things -- I got rid of the streching!
x = P(1,:);
y = P(2,:);
plot(x,y,"*")
### allow some streching
clear
A = [1 -2; 1 3;]
[a,b]=eig(A)
b = [1; 1]
P = [b]
for i=1:200 b = [1/sqrt(5)+0.01 0; 0 1/sqrt(5)+0.001]*A*b; P = [P, b]; end;
### I added a correcting factor!
### the matrix A streches things -- I got rid of the streching!
x = P(1,:);
y = P(2,:);
plot(x,y,"*")
|
X = [1 3*cos(.88); 1 2.3*cos(1.1); 1 1.65*cos(1.42); 1 1.25*cos(1.77); 1 1.01*cos(2.14)] Xt = transpose(X) y = [3.00; 2.30; 1.65; 1.25; 1.01] A = Xt*X b = Xt*y A\b 1.45/(1-0.811*cos(4.6)) |
Course Information |
Instructor: Adam Van Tuyl
Office: RB 2015
Office Hours: T-Th 10:15-11:30
Email: avantuyl AT lakeheadu.ca
Place and Time:
Class: RB 2047 TTh 8:30-10:00
Final Exam:
April 13 from 9AM-12PM in RC 0005
Textbook:
Linear Algebra and its applications (4th Edition)
by David C. Lay
Homework Soutions |
All the solutions to the homework and tests are posted electronically on
of Lakehead's library. The solutions are PDF files.
Homework Assignments |
Homework is given out every Thursday, and will be due, at the beginning of class, the following Thursday. Assignments must conform to the guidelines in the course outline. Assignments are posted below.
Assignment 1 (Due: Jan. 19)
Sec. 5.1 -- 6, 8, 12, 18, 22ab
Sec. 5.2 -- 8, 10, 18, 22, 24
Sec. 5.3 -- 2,4
Assignment 2 (Due: Jan. 26)
Sec. 5.3 -- 8, 14, 20, 22, 24, 28
Sec. 5.4 -- 2, 4, 8, 12, 16, 22
Assignment 3 (Due: Feb. 2)
Sec. 5.5 -- 4, 10, 16, 22
Sec. 5.6 -- 2, 8, 12, 16
Sec. 5.8 -- 2, 6, 8
Assignment 4 (Due: Feb. 16)
Sec. 6.1 -- 6, 12, 18, 20, 28
Sec. 6.2 -- 2, 10, 14, 20 ,24, 26, 32
Assignment 5 (Due: March 1)
Sec. 6.3 -- 2, 6, 8, 12, 18, 22
Sec. 6.4 -- 4, 8 , 10, 14, 18
Assignment 6 (Due: March 8)
Sec. 6.5 -- 4, 8, 12, 16, 18
Sec. 6.6 -- 2, 8 (do not need to graph), 12
Sec. 6.7 -- 4, 6, 8, 20
Assignment 7 (Due: March 22)
Sec. 7.1 -- 10, 14, 22, 26, 30
Sec. 7.2 -- 2, 6, 8, 22abc
Assignment 8 (Due: March 29)
Sec. 7.2 -- 22de, 23, 24
Sec. 7.3 -- 2, 4, 8, 10
On a separate sheet, make one exam appropriate question with answer.
Assignment 9 (Due: April 3) [NOTE DATE!]
Sec. 7.4 -- 4, 6, 10, 16, Bonus 24
Handouts |
All class handouts are available as PDF files.
Course Information
Course handout from first day of class
Midterm 1 Info Sheet
Review Sheet for first midterm
Midterm 2 Info Sheet
Review Sheet for second midterm
Final Exam Info Sheet
Review Sheet for Final Exam
Challenge Assignment |
Challenge 1
Due Feb. 2, 2012
Challenge 2
Due March 8, 2012
Grading Scheme |
Homework = 10%
2 Midterms = 50% (25% each)
Final Exam = 40%
Important Dates |
Jan. 6, 2012 - First Day of Classes
Feb. 9, 2012 - Midterm 1
Feb. 20-24, 2012 - Reading Break
March 7, 2012 - Last Day to Drop
March 15, 2012 - Midterm 2
April 5, 2012 - Last Day of Classes
April 10-20, 2012 - Exams
Links |