Math 307: Problems for section 1.3
1. Write down the vector approximating f (x) at interior points, the vector approximating
xf (x) at interior points, and the nite dierence matrix equation for the nite dierence
approximation with N = 4 for the dierential
Math 307: Problems for section 1.1
1. Use Gaussian elimination
1 2 3
1 2 3
(a) A =
5 6 7
5 6 7
to nd the solution(s) to Ax = b where
1 1
1
1
1
4
1 1 1 1
4
b = 1 ,
(b) A =
1 1 0
1
0
8
0 0
1
1
1
8
3
1
b = .
0
1
The process of Gaussian elimination i
Math 307: Problems for section 2.1
February 2, 2009
0
0
1
1
1
2 0 1 0 4
1. Are the vectors 1, 2, 3 , 2, 9 linearly independent? You may use MAT
2 1 2 0 7
3
1
0
1
1
LAB/Octave to perform calculations, but explain your answer.
Put the vectors in the
Math 307: Problems for section 4.5
December 20, 2009
1. Suppose that there is a xed population of cola drinkers each with a favourite among
Coke, Pepsi and Thums Up. Every month 3% of the Coke drinkers switch to Pepsi while
5% switch to Thums Up. Every mo
Math 307: Problems for section 1.3
1. Write down the vector approximating f (x) at interior points, the vector approximating
xf (x) at interior points, and the nite dierence matrix equation for the nite dierence
approximation with N = 4 for the dierential
Math 307: Problems for section 2.1
October 4, 2009
1
1
1
0
0
2 0 1 0 4
1. Are the vectors 1, 2, 3 , 2, 9 linearly independent? You may use MAT
2 1 2 0 7
1
1
0
1
3
LAB/Octave to perform calculations, but explain your answer.
Put the vectors in the c
Math 307: Problems for section 1.1
February 2, 2009
1. Use Gaussian elimination to nd the solution(s) to Ax = b where
11
1
1
1
1234
1 1 1 1
1
1 2 3 4
(b) A =
(a) A =
1 1 0
5 6 7 8 b = 1 ,
0
00
1
1
1
5 6 7 8
3
1
b = .
0
1
The process of Gaussian elim
Math 307: Problems for section 1.3
January 15, 2009
1. Derive the matrix equation to solve in order to nd the nite dierence approximation
with n = 4 for the dierential equation
f (x) + xf (x) = 0
subject to
f (1) = 1,
f (3) = 1.
We look for approximations
Math 307: Problems for section 2.3
October 19, 2009
1. Let D be the incidence matrix in the example done in the course notes.
1 1
0
0
0 1 1
0
0 1 1
D= 0
0 1 0
1
1
0
0 1
Using MATLAB/Octave (or otherwise) compute rref (D) and nd the bases for N (D),
R(D
Math 307: Problems for section 2.3
March 2, 2009
1. Let D be the incidence matrix in the example done in the course notes.
1 1
0
0
0 1 1
0
0
0 1 1
D=
0 1 0
1
1
0
0 1
Using MATLAB/Octave (or otherwise) compute rref (D) and nd the bases for N (D),
R(D) a
Math 307: Problems for section 3.2
November 9, 2010
1. Show that any 2 2 orthogonal matrix is either a rotation matrix or a reection matrix.
To obtain all possible 2 2 orthogonal matrices, we need to nd all possible orthonormal bases to use
as their colum
Math 307: Problems for section 3.3
November 3, 2009
1. Review of complex numbers:
(a) Show that |zw| = |z|w| for any complex numbers z and w.
(b) Show that zw = z w for any complex numbers z and w.
(c) Show that z z = |z|2 for every complex number z.
(a)
Math 307: Problems for section 1.2
February 2, 2009
Many problems in this homework make use of a few MATLAB/Octave .m les that are provided on
the website. In order to use them, make sure that the les are in the same directory that you are running
MATLAB/
Math 307: Problems for section 3.4
November 19, 2009
1. Calculate the Fourier coecients (cn s, an s and bn s) for the triangle function
2t
if 0 t 1/2
2 2t if 1/2 t 1
f (t) =
and show that the Fourier series decomposition of f (t) may be written
f (t) =
1
Math 307: Problems for section 2.3
October 19, 2009
1. Let D be the incidence matrix in the example done in the course notes.
1 1
0
0
0 1 1
0
0
0 1 1
D=
0 1 0
1
1
0
0 1
Using MATLAB/Octave (or otherwise) compute rref (D) and nd the bases for N (D),
R(D
Math 307: Problems for section 3.1
November 2, 2010
1. Use the CauchySchwarz inequality for real vectors to show
2
x+y
( x + y )
2
Under what circumstances is the inequality an equality?
x + y, x + y = x, x + x, y + y, x + y, y (linearity of the inner pro
Math 307: Problems for section 4.14.2
March 17, 2009
1. For the following matrices nd
(a) all eigenvalues
(b) linearly independent eigenvectors for each eigenvalue
(c) the algebraic and geometric multiplicity for each eigenvalue
and state whether the matr
Math 307: Problems for section 1.2
Many problems in this homework make use of a few MATLAB/Octave .m les that are provided on
the website. In order to use them, make sure that the les are in the same directory that you are running
MATLAB/Octave from (to s
function [N,R] = spaces307(A)
U=rref(A);
m = size(A,1); %extracts number of rows of A
n = size(A,2); %extracts number of columns of A
pivot_row = 0; 0lagging pivot row
0 x 1 vector having j-th entry 1 or 0 depending whether j-th column
% is a pivot column
%putting some points x_i into a vector X
%and the corresponding pts y_i into a vector Y
X=[0 0.2 0.4 0.6 0.8 1.0]
Y=[1 1.1 1.3 0.8 0.4 1.0]
%command plot(X,Y) will plot pts (x_i,y_i) joined by straight lines
%adding a third argument "o" plots pts as littl
%This is the editor window. I can write code here and run it in the main
%window by hitting the play (green triangle) button above or by typing the
0ame of the file (in this case matrixdemo) in the command window
fprintf('When the code stops it is on a pa
I.1.5. Norms for a vector
Norms are a way of measuring the size of a vector. They are important when we study how vectors
change, or want to know how close one vector is to another. A vector may have many components
and it might happen that some are big a
%Let's generate the matrix L directly for our 3D triangular prism. Since
%the resitances are all 1, the diagonal entries will be the sum of the
%edges connecting each node and the off diagonal elements will be either -1
%if the nodes are connected or zero
UBC MATH 307, 2015 Summer 1: Applied Linear Algebra
Description: This course is organized around a collection of interesting applications.
Examples include: interpolation, finite difference approximations, formula matrix of
a chemical system, least square
%Let's look at a quick example of some vectors to show the nonuniqueness of
%linear independence if not all the vectors are linearly independent
%Start with a matrix with the following 5 vectors in R3
A=[1;2;3] [1;1;1] [4;5;6] [7;8;8] [2;2;2]
pause;
%If w
function plotcubic2(xl,xu,yl,yu,zl,zu)
%
% plots the cubic polynomial in the interval [xl,xu]
% whose values (resp. 2nd derivatives) at the endpoints
% are yl,yu (resp. zl,zu)
%
% The code contains expressions that appear to add a number to a vector, whic
%This is a demo used to solve the equation f'=1 that we talked about in
%class. We will also look at the matrix L with the Dirichlet conditions
close all
clear all
%Let's set N=50 with the idea that this is enough to have the appearance of
%a continuous s
close all
clear all
%Let's load the signal
y=loadaudio('F6.baroque','au',8);
% this is command for Octave only
% for MATLAB the command is y=auread('F6.baroque.au');
%The sampling rate is 22050 samples/s
Fs=22050;
%N is the size of this vector, let's get
close all
clear all
%Let's build an internet with 5 sites
%(a) blogger
%(b) small town newspaper
%(c) big city newspaper
%(d) national newspaper
%(e) conspiracy nut
%Let's have a->b,c,d; b->c,d; c->a,b,d; d->a,c; e->none
%Let's build P. It has 1/outgoing