Systems of Linear Equations and System Equivalency

Systems of Linear Equations (and System Equivalency) [Video]

I begin with 2 by 2 and 3 by 3 linear systems of equations. I then discuss consistent and inconsistent systems of linear equations and work through several examples. The general system of linear equations is defined, and parametrizing a solution set of a linear system is demonstrated. I explain augmented matrices, row operations, and when two linear systems of equations are called equivalent.

Canonical Forms and Jordan Blocks Stack of Wooden Blocks with binary code on White Backgroun scaled 1

Canonical Forms and Jordan Blocks

Are you trying to understand what a Jordan basis is? In this article, I go over Jordan’s Theorem and the Jordan Canonical Form. I work through several examples.

Diagonalization of a Matrix fluid dynamics amplitude reaches a critical level start to occur that causes large amounts of wave energy to be transformed into turbulent kinetic energy scaled 1

Diagonalization of a Matrix (with Examples)

Do you know the Diagonalization Theorem? In this article, I cover this theorem, two corollaries, and work through many examples. The method of diagonalization is illustrated in the following examples.

Eigenvalues and Eigenvectors Linear Transformation Matrix and Invertibility Error basically code on screen of the monitor scaled 1

Eigenvalues and Eigenvectors (Find and Use Them)

Are you looking for more information on eigenvalues and eigenvectors? In this article, I work through several examples and go over the characteristic equation, and then I go over even more examples. All theorems are provided with proof.

Determinant of a Matrix Example of calculating of the determinant of a given two by two matrix scaled 1

The Determinant of a Matrix (Theory and Examples)

Okay, so there is the definition of a determinant, and there are ways of computing the determinant. The definition of determinant involves patterns and a signature, and this approach is useful in proofs. But to determine the determinant in practice, it’s helpful to have expansion by cofactor. In this article, I detail both.

Inner Products and Orthonormal Bases Face detection and recognition of man Computer vision and machine learning concept scaled 1

Inner Products and Orthonormal Bases

You have been familiar with the Cartesian plane for a while, and so you know about angles. How do we work with angles in a vector-space? In this article, I go over inner-product spaces and work through the Cauchy-Schwarz Theorem and the Triangular Theorem. I also discuss orthonormal bases.

Gram Schmidt Process Global Information Network Protection and storage of digital data using the blockchain technology Artificial intelligence based on neural networks scaled 1

Gram-Schmidt Process and QR Factorization

In this article, I explain what the Gram-Schmidt Process is and what a QR Factorization is. I do this by working through several examples. Also, I include the commutative diagrams for the corresponding transformations.

Orthogonal Matrix and Orthogonal Projection Matrix Vasco da Gama Bridge in Lisbon Portugal night scene scaled 1

Orthogonal Matrix and Orthogonal Projection Matrix

In this article, I cover orthogonal transformations in detail. After that, I present orthogonal and transpose properties and orthogonal matrices. Towards the end, I examine the orthogonal projection matrix and provide many examples and exercises.

Orthonormal Bases and Orthogonal Projections Four projections top front orthogonal and perspective of a blank beverage can scaled 1

Orthonormal Bases and Orthogonal Projections

In this article, I begin with the fact that orthonormal vectors are linearly independent and thus form a basis for the subspaces generated by them. After that, I explore the orthogonal projection and properties of the orthogonal complement. Towards the end, I detail the Pythagorean Theorem, the Cauchy-Schwarz Theorem, the Law of Cosines, and the Triangular Inequality Theorem.