Felix Bloch talking to Heisenberg on a walk
Bloch: Space is simply the field of linear operators.
Heisenberg: Nonsense, space is blue and birds fly through it.

Linear Algebra

This is a very short introduction. Linear algebra is vast and you can get as much accurate as you want in definitions, proofs, and so on. That is not what we want to do here. You will learn the basic definitions and operations on vectors and matrices. These basics concepts are enough to get started with quantum computing.

Vector

A vector is an ordered set of numbers. For example \(\bm{v} = (-1, 0.5, 3)\) is a vector.¹ It contains three numbers which are also called the elements. A vector can have any number of elements. A vector with only one element is just a scalar (just a number).

Column vs Row

There are two ways to list the elements of a vector. In a row

\[\bm{v} = \begin{pmatrix} -1 & 0.5 & 3 \end{pmatrix}\]

or in a column

\[\bm{u} = \begin{pmatrix} -1 \\ 0.5 \\ 3 \end{pmatrix}\]

The row vector \(\bm{v}\) and the column vector \(\bm{u}\) are considered to be different. That is why I used two different letters to represent them. There is an operation that relates these two to each other. It’s called transpose and is shown by T in the superscript.

\[\bm{v}^T = \bm{u} \\ \bm{u}^T = \bm{v} \\\]

The transpose of a column vector is a row vector and vice versa. If you transpose a vector twice, you get the same vector.

\[(\bm{v}^T)^T = \bm{v}\]

Scalar multiplication

You can multiply a vector by a scalar (which is just a number). For that, you would just need to multiply individual elements of the vector by that same scalar.

\[a\bm{u} = \begin{pmatrix} -a \\ 0.5a \\ 3a \end{pmatrix}\]

Footnote

1. A boldface variable \(\bm{v}\) is usually used to denote a vector. Another way of denoting a vector is to put an arrow on top \(\overrightarrow{v}\). I’m going to stick to the boldface notation. ↩