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Linear Algebra Tutorial

By Kardi Teknomo, PhD.

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In this simple Linear Algebra tutorial, you will review the basic operations on vectors, matrices, solving linear equations and eigen systems. I made free educational interactive programs online for almost every topic. The programs are well suited to guide discovery experiences where you can focus on the concept of linear algebra and let the program compute the more complex calculation. My focus is to introduce linear algebra with clarity of explanation. Thus, I try to avoid mathematical detail and you should read the proof of all the properties and theorems from the references. Manual examples on how to compute are given. Random examples of the interactive programs are useful to give you theoretically infinitely many examples. No download file is necessary for this tutorial and you must set JavaScript enabled in your browser.

First, take a look on how to input a vector or a matrix in the interactive program, and then you can select the topics below that you are not familiar or you want to review.

If you are a first time learners, who are not familiar with most of the topics, you should browse each section & each topic of the tutorial in sequence. You should also try the random example of the interactive programs to gain your understanding of the concept. You may want to skip the properties and notes below the interactive program at first reading. After you are familiar with the name, definition and operations, you will find that the notes and properties listed are useful for second reading or whenever you need to come back to the particular topics.

The strength of linear algebra lies on the properties of each operation. I try to collect these properties on the same page of the context so that it is easier for you to find them back. Thus, you will find some of the properties are repeated in different pages for the appropriate context.

Here are the topics of this linear algebra tutorial:

Matrix & Vector Size & Validation (interactive program)
Vector Algebra
What is Vector? (interactive program)
Vector Norm (interactive program)
Unit Vector (interactive program)
Vector Addition (interactive program)
Vector Subtraction (interactive program)
Vector Scalar Multiple (interactive program)
Vector Multiplication
Vector Inner Product (interactive program)
Vector Outer Product (interactive program)
Vector Cross Product (interactive program)
Vector Triple Cross Product (interactive program)
Vector Triple Dot Product (interactive program)
Scalar Triple Product (interactive program)
Orthogonal & Orthonormal Vector (interactive program)
Cos Angle of Vectors (interactive program)
Vector Projection (interactive program)
Matrix Algebra
What is a matrix and why do we need matrix?
Special Matrices
Matrix One (interactive program)
Null Matrix (interactive program)
Matrix Diagonal (interactive program)
Is Diagonal Matrix? (interactive program)
Identity Matrix (interactive program)
Matrix Special Properties
Matrix Determinant (interactive program)
Matrix Sum (interactive program)
Matrix Trace (interactive program)
Matrix Basic Operation
Is Equal Matrix? (interactive program)
Matrix Transpose (interactive program)
Matrix Addition (interactive program)
Matrix Subtraction (interactive program)
Matrix Multiplication (interactive program)
Matrix Scalar Multiple (interactive program)
Matrix Element-wise product (Hadamard Product) (interactive program)
Matrix Horizontal Concatenation (interactive program)
Matrix Vertical Concatenation (interactive program)
Elementary Row Operations
Matrix RREF (interactive program)
Finding inverse using RREF (Gauss-Jordan) (interactive program)
Finding Matrix Rank using RREF (interactive program)
Matrix Inverse (interactive program)
Is Singular Matrix? (interactive program)
Linear Transformation
Matrix Generalized Inverse (Moore Penrose interactive program)
Solving System of Linear Equations (interactive program)
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent (interactive program)
Change of basis
Matrix Rank (interactive program)
Matrix Range (interactive program)
Matrix Nullity & Null Space (interactive program)
Eigen System
Matrix Eigen Value & Eigen Vector
Symmetric Matrix (interactive program)
Matrix Eigen Value & Eigen Vector for Symmetric Matrix (interactive program)
Similarity Transformation and Matrix Diagonalization
Matrix Power
Orthogonal Matrix (interactive program)
Spectral Decomposition (interactive program)
Singular Value Decomposition (interactive program)
Resources on Linear Algebra

See Also: Regression, Distance, Similarity, Sum

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This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\


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