A few years ago, I remember being faced with an introductory physics problem. It was the first time I had encountered the idea of acceleration, and I was puzzled by the notation for it.

This is a continuation of my Linear Algebra series, which should be viewed as an extra resource while going along with Gilbert Strang’s class 18.06 on OCW. This can be closely matched to Lecture 8 in his series.

Here, we continue our discussion about solving linear systems with elimination. In my Gaussian Elimination series, we explored how square, invertible matrices can be solved by method of elimination and row exchanges — but we never delved into solving rectangular, non-invertible systems.

In the last lesson, we explored how non-square systems can be solved by using Gaussian elimination. …

This is a continuation of my Linear Algebra series, which should be viewed as an extra resource while going along with Gilbert Strang’s class 18.06 on OCW.

Let’s introduce a small additional concept in linear algebra — this is the Reduced Row Echelon Form of a matrix. I’d highly recommend reading the article before this, which details how to solve Ax = 0 in general.

Leaving right where we left off, we eliminated through our A matrix to get something in echelon form, symbolized by the variable U. …

This article assumes a base knowledge of all basic matrix and vector manipulation concepts, like dot products, linear combinations, as well as ideas involving span, vector spaces and subspaces, and column spaces.

Review of Null Space

Continuing off the last article, we introduced ideas of the null space.

To review, the null space is the vector space of some group of x that satisfy Ax = 0. …

Now that we have explored the column space, we can explore the other vector subspace that matrices can offer, which is the null space. First, we must be familiar with the matrix way of representing a system of equations. I cover this in the first part of my gaussian elimination tutorial.

The entire concept of null space is rather simple. We are trying to find all possible x in a matrix equation Ax = 0. A combination of all these x will form their own subspace, this time in R^n. Let’s walk through all of that .

When do we have a null space?

Let’s start off with an example, in the language of (3, 3) matrices. Let’s also put these in Ax = 0 form, since we’re solving for x. …

This article assumes knowledge of Gaussian Elimination, matrix multiplication, linear combinations, and a few other concepts. If Gaussian Elimination seems foreign, I have a series of three articles walking through the meanings and mechanics of it.

Now that we have fundamentally covered the elimination way of looking at a system of equations Ax = b, we can delve a little more into the abstract concepts of linear algebra that underlie the system.

We can analyze the rows and columns of our matrix A to understand the concept of vector spaces and subspaces.

What is a Vector Space?

A vector space, and later, subspaces, are a group of vectors that are closed under scalar multiplication and addition. …

Matrix transposes and symmetric matrices are linked — in fact, the definition of a symmetric matrix is that a symmetric matrix A’s transpose gives back the same matrix A.

This is a continuation of my linear algebra series, tied with the 18.06 MIT OCW Gilbert Strang course on introductory linear algebra. Let’s jump right in with the basics on transposes.

Transposing a matrix

We transpose a two-dimensional matrix by using it’s rows as columns, or inversely, as it’s columns as rows. That is all it takes to do the simple operation.

In the last article, we discussed how we could express Gaussian Elimination, in terms of matrix multiplications, and how we could multiply those elementary matrices into one single matrix E, which would bring A to the upper triangular U.

Let’s jump right in.

The Problem With Elementary Matrix E

Let’s take a look again at what our example matrix E (our multiple of our three individual elimination matrices) looked like in the last article.