Building matrices in C++

This section draws heavily on the textbook Linear Algebra Done Right by Sheldon Axler

In the previous issue I didn't go that deeply into the mathematical defintion behind a vector. Sadly, before we can truly understand what a matrix is, we must first understand what a vector is.

In grade school, you should have learned about \(\mathbb{Q}\), the rational numbers, \(\mathbb{Z}\), the integers, and most importantly for this discussion, \(\mathbb{R}\), the set of all real numbers. The set of the real numbers(\(\mathbb{R}\)) include the rational numbers, which are all numbers that can be represented in the form \(\frac{p}{q}\), and the irrational numbers, all of the numbers that are too cool for school, like \(\pi\), \(\varphi\), and \(\sqrt{2}\). It does not include, however, what we call imaginary numbers: numbers that include an imaginary part, \(i\), the imaginary unit, which is defined as the $\sqrt{-1}. Imaginary, because it's not a real number, and unit, because it serves the same purpose on the imaginary axis as 1 on the real axis. A complex number, is an ordered pair \((a, b)\), where \(a, b \in \mathbb{R}\), and written as \(a + bi\), where \(i\) is again that imaginary part. We denote the set of all complex numbers with \(\mathbb{C}\). We define \(\mathbb{F}\) as either \(\mathbb{R}\) or \(\mathbb{C}\).