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Identity Matrix and Inverse Matrix

We introduce the inverse matrix and the identity matrix. In addition, we learn how to solve systems of linear equations using the inverse matrix. The identity matrix is a matrix in which the diagonal entries are 1, and all other entries are zero. It is a more restrictive form of the diagonal matrix. It

What is a Diagonal Matrix

We introduce and discuss the applications and properties of the diagonal matrix, the upper triangular matrix, and the lower triangular matrix. A diagonal matrix is a square matrix in which all entries are zero, except for those on the leading diagonal. It is also called the scaling matrix because multiplication with the diagonal matrix

Gaussian Elimination and Gauss Jordan Elimination: An Introduction

We introduce Gaussian elimination and Gauss Jordan Elimination, more commonly known as the elimination method, and learn to use these methods to solve linear equations with several unknown variables. We also introduce the row echelon form of a matrix and discuss the difference between Gaussian elimination and Gauss Jordan elimination. Gaussian Elimination Method Gaussian

Understanding the Determinant of a Matrix

We introduce the matrix determinant and its uses. Furthermore, we learn how the determinant provides us with a shortcut to finding the inverse matrix. What is the matrix determinant? The determinant is a scalar value that can be obtained from a square matrix and which can be used to find the inverse of a

How to Use the Transformation Matrix

We learn how to construct transformation matrices and how to use them to rotate, stretch or otherwise transform vectors. A transformation matrix scales, shears, rotates, moves, or otherwise transforms the default coordinate system. In the process it maps coordinates from the current coordinate system to the one that resulted out of the transformation. A

How to Perform Matrix Multiplication

In this post we learn how to perform matrix multiplication and why we need to be mindful of matrix dimensions. Furthermore, we look at the properties of matrix multiplication. Matrix multiplication is an operation that consists of the element-wise multiplication of all entries in a row of the first matrix with all entries in

How to Perform Basic Matrix Operations

We introduce and define matrices. Furthermore, we learn how to perform matrix addition and matrix subtraction. A matrix is simply a 2-dimensional vector. While vectors only have one dimension m, matrices have 2 dimensions m and n We say A is an matrix. As you can see, we usually denote matrices with capital letters.

What is Linear Independence: An Explanation by Example

In this post we define linear independence and walk through an example to develop an intuitive understanding of the concept. What is Linear Independence? When a set of several vectors is linearly independent, it is not possible to represent one vector as a linear combination of the remaining vectors in the set. Two orthogonal vectors

Dot Product and Orthogonal Vectors

In this post, we learn how to calculate the dot product between two vectors. Furthermore, we look at orthogonal vectors and see how they relate to the dot product. What is the Inner Product? The inner product of two vectors is the sum of the element-wise products of two vectors. The result of the

Unit Vector: Background and Definition

In this short post, we will learn how to calculate the magnitude of a vector and how to obtain the unit vector. The magnitude of a Vector A vector has a direction and a magnitude (or length). The following vector has a direction of 1 on the x-axis and 3 on the y-axis.  But