A calculus of the absurd

Chapter 18 Linear algebra

18.1 Properties of matrices

On the off-chance that someone other than me reads these, please don’t read this section in one go; it’s here for reference.

18.1.1 Introduction

A matrix has \(m\) rows and \(n\) columns. For example, a \(2 \times 3\) matrix would look something like this

\begin{equation} \begin{pmatrix} 0 & 1 & 2 \\ 3 & 5 & 8 \end {pmatrix} \end{equation}

Of course, the values in a matrix can be anything. If we want to denote a matrix having only real numbered elements, we can write this as \(\mathbb {R}^{m \times n}\) (we can also do the same for other sets, for example \(\mathbb {N}^{m \times n}\) for an \(m\) by \(n\) matrix whose elements are all natural numbers).

We can also define a matrix using the notation

\begin{equation} (f(i, j))_{ij} \end{equation}

This means that the \(j\)th column of the \(i\)th row is equal to \(f(i, j)\).