Matrix Algebra: Vectors

Many university STEM major programs have reduced the credit hours for a course in Matrix Algebra or have simply dropped the course from their curriculum.   The content of Matrix Algebra in many cases is taught just in time where needed.  This approach can leave a student with many conceptual holes in the required knowledge of matrix algebra.

In this series of blogs, we bring to you ten topics that are of immediate and intermediate interest for Matrix Algebra.

Here is the second topic where we talk about vectors, binary operations on vectors, set up the concept of linear combination and linear independence of vector.  Learn how vectors are used to set up simultaneous linear equations.  Get all the resources in form of textbook content, lecture videos, multiple choice test, problem set, and PowerPoint presentation.

Vectors


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Matrix Algebra: Binary Operations

Many university STEM major programs have reduced the credit hours for a course in Matrix Algebra or have simply dropped the course from their curriculum.   The content of Matrix Algebra in many cases is taught just in time where needed.  This approach can leave a student with many conceptual holes in the required knowledge of matrix algebra.

In this series of blogs, we bring to you ten topics that are of immediate and intermediate interest for Matrix Algebra.

Here is the third topic where we talk about binary operations of matrices – subtraction, addition, and multiplication.  Linear combination of matrices and rules of binary operations are discussed. Get all the resources in form of textbook content, lecture videos, multiple choice test, problem set, and PowerPoint presentation.

Binary Operations


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Matrix Algebra: Introduction

Many university STEM major programs have reduced the credit hours for a course in Matrix Algebra or have simply dropped the course from their curriculum.   The content of Matrix Algebra in many cases is taught just in time where needed.  This approach can leave a student with many conceptual holes in the required knowledge of matrix algebra.

In this series of blogs, we bring to you ten topics that are of immediate and intermediate interest for Matrix Algebra.

Here is the first topic where we define a matrix, vector, submatrix, square matrix, triangular matrix (upper and lower), diagonal matrix, identity matrix, and diagonally dominant matrix.  Get the information in form of textbook content, lecture videos, multiple choice test, problem set and PowerPoint presentation.

Introduction 


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How much computational time does it take to find the inverse of a square matrix using Gauss Jordan method?  Part 1 of 2.

Problem Statement

How much computational time does it take to find the inverse of a square matrix using Gauss Jordan method?  Part 1 of 2.

Solution

To understand the solution, you should be familiar with the Gauss Jordan method of finding the inverse of a square matrix.  Peter Young of UCSC describes it briefly in this pdf file while if you like watching an example via a video, you can see PatrickJMT doing so.  You also need to read a previous blog where we calculated the computational time needed for the forward elimination steps on a square matrix in the Naïve Gauss elimination method.   We are now ready to estimate the computational time required for Gauss Jordan method of finding the inverse of a square matrix.

GJ Inverse Blog

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Clearing up the confusion about diagonally dominant matrices – Part 3

You can view the above document as a pdf file as well.

Other blogs on diagonally dominant matrices
Clearing up the confusion about diagonally dominant matrices – Part 1

Clearing up the confusion about diagonally dominant matrices – Part 2

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Clearing up the confusion about diagonally dominant matrices – Part 2

In a previous post, we discussed the confusion about the definition and associated properties of diagonally dominant matrices.  In this blog, we answer the next question.

What is a weak diagonally dominant matrix?

The answer is simple – the definition of a weak(ly) diagonally dominant matrix is identical to that of a diagonally dominant matrix as the inequality used for the check is a weak inequality of greater than or equal to (≥).  See the previous post on Clearing up the confusion about diagonally dominant matrices – Part 1 where we define a diagonally dominant matrix.

Other blogs on diagonally dominant matrices
Clearing up the confusion about diagonally dominant matrices – Part 1

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MOOC on Introduction to Matrix Algebra released

Introduction to Matrix Algebra is available as a MOOC now on Udemy. Of course, it is free! https://www.udemy.com/matrixalgebra/.  You will have a lifetime access to 177 lectures, 14+ hours of high quality content, 10 textbook chapters complete with multiple choice questions and their complete solutions.

Learning Objectives are

  • know vectors and their linear combinations and dot products
  • know why we need matrix algebra and differentiate between various special matrices
  • carry unary operations on matrices
  • carry binary operations on matrices
  • differentiate between inconsistent and consistent system of linear equations via finding rank of matrices
  • differentiate between system of equations that have unique and infinite solutions
  • use Gaussian elimination methods to find solution to a system of equations
  • use LU decomposition to find solution to system of equations and know when to choose the method over Gaussain elimination
  • use Gauss-Seidel method to solve a system of equations iteratively
  • find quantitatively how adequate your solution is through the concept of condition numbers
  • find eigenvectors and eigenvalues of a square matrix

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