MTH 719 Applied Linear Algebra
Lab 7
LU Factorization
To nd the LU factorization of a matrix A in Matlab we use the lu command. For example, it
we type
> [L U P] = lu(A)
we get the full LU factorization of A such that LU = P A, where P is a permutation ma
MTH 719 Applied Linear Algebra
Lab 1 Solutions
Exercise 1
True or False: (Give a proof or a counterexample)
These are all false as discussed in class. The main idea is to nd a single counterexample that
is as simple as possible.
If the number of equation
http:/www.rnet.ryerson.ca/~rphan/mth719/w2013
Raymond Phan
Ph.D. Candidate
Department of Electrical & Computer Engineering
Ryerson University
Topics Covered
Introduction
Quick intro to me and contact information
Well be using MATLAB in the course. Yay?
MTH 719 Applied Linear Algebra
Lab 8
This week I have published two test functions to evaluate your programs for Assignment 2. There
are two versions of your test function because samll dierences in the way you may have written
your code will eect the res
Mathematics 719 Applied Linear Algebra
February 19, 2015
Midterm Test 2
You are allowed to use any Matlab resources, including any m-les you have on a USB stick. Otherwise you
may not use any other resources on the computer, including, but not limited to
MTH 719 Applied Linear Algebra
Lab 8 Solutions
Exercise 1
Find a nonsingular matrix P such that P A = EA , where
1 2 3 4
A= 2 4 6 7
1 2 3 6
We may perform gauss elimination on the augmented matrix (A|I) which will lead us to
(EA |P ).
In Matlab
A = [1 2
MTH 719 Applied Linear Algebra
Lab 7 Solutions
Exercise 1
In Lab 6 we considered the three matrix norms |A|1 , |A|2 and |A| . Write a Matlab function
x = oneNorm(A) where A is a matrix and x = |A|1 . Do this without using Matlabs norm function.
Do the sam
MTH 719 Applied Linear Algebra
Lab 6
Norms and Conditioning Numbers
For a vector X Rn we have dened three norms,
|X|1
= |x1 | + |x2 | + + |xn |
|X|2
=
|X|
=
|x1 |2 + |x2 |2 + + |xn |2
maxcfw_|x1 |, |x2 |, , |xn |
Given a vector X in Matlab, we may nd the
Mathematics 719 Applied Linear Algebra
February 12, 2015
Midterm Test 1
You are allowed to use any Matlab resources, including any m-les you have on a USB stick. Otherwise you
may not use any other resources on the computer, including, but not limited to
MTH 719 Applied Linear Algebra
Lab 4 Solutions
Exercise 1 Construct a homogeneous system of three equations in four unknowns that has
2
3
1
0
x2
0 + x4 2
0
1
as its general solution. What is the rank of the coecient matrix of your system?
Before do
MTH 719 Applied Linear Algebra
Lab 10 Solutions
Exercise 1
Let
51
144
A=
111
102
18 16 14
48 50 41
30
52
36
24 54 35
Find A. That is nd all matrices B such that B 2 = A.
The matrix A is diagonalizable, i.e., A = P DP 1 , where P is an invertible matri
MTH 719 Applied Linear Algebra
Lab 12
QR factorization
Let A be an m n matrix with rank n. Thus m n and the columns of A are linearly
independent. In this case we may nd the reduced and the full QR-factorization of A.
A reduced QR-factorization is an m n
MTH 719 Applied Linear Algebra
Lab 12 Solutions
Exercise 1
We should like to design matrices with various properties. Often this is easy to do with diagonal
matrices. However, we wish to create interesting matrices with the desired properties. Let us agre
MTH 719 Applied Linear Algebra
Lab 3
Sparse Matrices
Often matrices have well placed zeros, which makes them easier to work withtheoretically at
least. We already know that in principle it takes one or two orders of magnitude less ops to solve a
triangula
MTH 719 Applied Linear Algebra
Lab 6 Solutions
Exercise 1
Recalling the theorems we proved in class for calculating matrix norms |A|1 , |A|2 and |A| ,
write Matlab functions of the form N = myNorm(A) that calculate these norms without using
Matlabs norm f
MTH 719 Applied Linear Algebra
Lab 10
Standard Inner Product
Given two vectors x and y in Rn their standard inner product is dened to be the number
T
x y = x1 y1 + x2 y2 + + xn yn . This is also called the dot product, and many authors use the
notation x
MTH 719 Winter 2013 Final Exam
1. F
2. F
3. F
4. T
5. F
6. T
7. F
8. T
9. F
10. T
11. F
12. F
True of False
1. F
2. F
3. F
4. T
5. F
6. F
7. F
8. T
9. T
10. T
MTH 719 Applied Linear Algebra
Lab 4
Function Handles
We would like to write functions in MATLAB of the form y = f (x). Consider the formula
y = x2 +1. It takes the input variable x and calculates the output y based on the formula y = x2 +1.
One way to de
MTH 719 Applied Linear Algebra
Lab 3 Solutions
Exercise 1
Recall from lecture that we may nd a numerical approximation to the two-point boundary value
problem
y (t) = 1, 0 t 1, y(0) = 0, y(1) = 0
by solving a tridiagonal system of equations. We divide [0
Mathematics 719 Applied Linear Algebra
Practice Midterm Questions
You are allowed to use any Matlab resources. Otherwise you may not use any other resources on the
computer, including, but not limited to the internet. No phones are allowed. You may not us
MTH 719 Applied Linear Algebra
Assignment 1
Due February 9th in class.
Include all written answers, tables, graphs, images and for each problem. Send a separate
.m file for each question where you are requested to do so. Also remember to send along
any au
MTH 719 Applied Linear Algebra
Assignment 1 Solutions
1. Given n data points
(x1 , y1 ), (x2 , y2 ), . . . (xn , yn ) with x1 < x2 < < xn ,
(1)
we may always find the unique (at most) n 1 degree polynomial, P (x), that interpolates these data, i.e., P (xj
MTH 719, Winter 2017
Lab 3
Underdetermined Systems of Equations I
An m n system of equations is called underdetermined if m < n, i.e., the system has less equations than unknowns. Typically (though not always) such systems have infinitely many solutions.
Mathematics 719 Applied Linear Algebra
Assignment 3
Due: April 13, 2017 in class.
Include all written answers, tables, graphs and images for each problem. Send a separate .m
file for each question as appropriate. Also remember to send along any auxiliary