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The Dem
Fundamental Subspaces
Defined by a Matrix
Math 443 Lecture
Based on ALA
CHAPTER TWO
An m x n Matrix Relates a Vector Space
of Dim n to a Vector Space of Dim m
Matrix A is a m x n
Vector X is a n x 1
Y= AX
Vector Y is a m x 1;
m x 1 = (m x n) .(n x 1)
Defi
MATH 443 October 18, 2016
OPTIONAL - MIDTERM I - Extra Credit Problem
Extra Credit Problem. A collection of N vectors in
2
are given by columns of the
2 N
matrix
P. Find a method that enable you to arrange the N vectors (given by columns of P) into THREE
MATH 443 October 18, 2016
OPTIONAL - MIDTERM I - Extra Credit Problem
Extra Credit Problem. A collection of N vectors in ! 2 are given by columns of the 2 N matrix
P. Find a method that enable you to arrange the N vectors (given by columns of P) into TH
MATH 443
Fall 2016
PRACTICE MIDTERM I
YOUR NAME:
In this Practice Midterm,
FIRST try to solve ALL Problems, as in the midterm in-class with
limited time 70 minutes.
NEXT PRACTICE MORE with ALTERNATE Problems
Time: 70 minutes work on problems + 5 minutes t
FALL 2016
Math 443 (A. Assadi)
HW #6 Matrix Algebra over Complex Numbers
BACKGROUND
Definition 1. Let A be a square matrix with entries in the field of complex numbers. The
conjugate transpose (also called Hermitian transpose) of the n n matrix
n n matrix
MATH 443 - Fall 2016
Practice MIDTERM I
Practice MIDTERM I
Solutions will be provided. There are additional problems to test yourself. These are called
Alternate Problem 1, Alternate Problem 2 , etc.
Notation. We have a matrix M, a matrix Q, and use M, Q
LU & Cholesky Decomposi2on
Overview:
1. Deni2on and Basic examples
2. DooliAle Algorithm
3. C+ Code
4. Cholesky Decomposi2on
5. Applica2ons
6. Matlab Useage
Deni2on.
Let A be a square matrix. An LU decomposi2on
MATH 443
Fall 2013
1st Week
An Introduction to MATLAB
Environment
What is MATLAB?
At first it was just a MATrix LABratory
Now: An object oriented high-performance language for technical
computing, visualization and programming
MATLAB environment overvi
Positive Definite Matrices
Definition 3.20
An nxn matrix K is called positive definite if it is symmetric, K T K ,
and satisfies the positivity condition
xT Kx 0 for all 0 x n
(3.43)
K 0 means K is a symmetric positive definite matrix.
Theorem 3.21
Every
Exercise With Arrays And Matrices
MATLAB provides functions for creating a number of standard arrays, such those containing all ones or all zeros.
Other standard arrays are identity matrices, arrays of random numbers, diagonal arrays, and arrays whose ele
COMPUTER LAB SESSION (Fall 2013 Math443 - A. Assadi)
Arithmetic Operation with MATLAB Commands
The basic arithmetic operations for numbers are the familiar ones: add is + , subtract is - ,
multiply is * and divide is / . Powers are indicated with ^ by typ
M ath-443 Informal Notes I
F all 201 3 - A. Assadi
M ATLAB Scripts And Functions
MATLAB has a powerful programming language as part of its core software. This language is high-level,
that is, its programs get interpreted into another language such as C, a
FALL 2013
Math 443 A. Assadi
The Inverse of a Matrix
Solved Examples and Exercises with Matrices
The main purpose of this note is to illustrate how simple mathematical proofs are written
down. We use properties of the inverse of a matrix as an example.
Wh
M ath-443 Informal Notes II
F all 201 3 - A. Assadi
M ATLAB Programs
Recall: MATLAB program file-names must be saved with the extension ".m", also called m-files.,
because of this obligatory file-name extension.
The simplest MATLAB programs are called scr
Definition of Inner Product
An inner product on the real vector space V is a pairing that takes two vectors
v, w V and produces a real number (v, w ) . The inner product is required
to satisfy the following three axioms for all u, v, w V , and scalars c,
COMPUTER LAB SESSION (Fall 2013 Math443 - A. Assadi)
Basic Tools of the Trade!
The purpose of these notes to help the students who need to review linear algebra for their
projects to develop the appropriate computational skills and practice their understa
The Closest Point
Let v1 ,., v n be a basis for V. The general element v V is a
linear combination of the basis vectors.
v x1v1 . xn vn Ax,
where A=(v1v 2 .v n ) is the m x n matrix formed by the
column basis vectors and x=(x1 ,x 2 ,.,x n )T are the
coord
Positive Definite Matrices
Definition 3.20
An nxn matrix K is called positive definite if it is symmetric, K T K ,
and satisfies the positivity condition
xT Kx 0 for all 0 x n
(3.43)
K 0 means K is a symmetric positive definite matrix.
Theorem 3.21
Every
Definition of Inner Product
An inner product on the real vector space V is a pairing that takes two vectors
v, w V and produces a real number (v, w ) . The inner product is required
to satisfy the following three axioms for all u, v, w V , and scalars c,
FALL 2016
Math 443 (A. Assadi)
Examples and Solved Exercises for Matrix Algebra
The Inverse of a Matrix and Matrix Equations
This note is an extended write-up of the previous notes on inverse of matrices. The objectives are to help
students for the follow