Introduction to Linear Algebra

Linear Algebra

If you have never heard of linear algebra, to put it simply, it is the study of everything made up of dots, lines and planes — these are the shapes which we call linear within our physical realm. Linear objects are fundamental in our universe and they are the building blocks of the digital world. For instance, the pixels on your screen are singular points in space, the graphics you see in games, are rendered by millions of triangles which are made up of lines and the recommendation algorithms on YouTube which suggests Penguin videos are partially a result of neural networks which is just a collection of linear equations.

Since linear objects are basic, we are able to learn so much about them and often times we can reduce nonlinear problems into linear ones and solve them. In fact, almost every wild function you can dream of can be written as a (infinite) linear sum of functions. There is a special name for this and if you have studied some maths, engineering or physics you may have encountered it before — it is known as the Fourier Series.

We will assume that you are familiar with basic set theoretic notation (I might write up a page later for that for people unfamiliar), the concept of real and complex numbers and elementary functions such as polynomials, trigonometric, logarithmic and exponential functions.

Before we fully dive into it, it is important to understand that maths is a subject in which we continuously try to generalise things further and further until they become really fundamental/abstract objects. This is particularly the case with algebra as a subject field, which may not be the algebra you are familiar with from high school. For instance, the vectors in which I will introduce in the beginning will soon be generalised in a more abstract setting, but we will use this setting early on to motivate why we define these abstract objects later on. As a side note, when I mentioned the term objects, it can refer to various things such as numbers, functions, matrices, differentials, tensors and so on.

Vectors

As I mentioned above, vectors can be many different objects, but the most fundamental object that we should start off with to motivate the rest of the theory is the Euclidean vector (for this section we will simply refer to Euclidean vectors as vectors unless stated otherwise). An Euclidean vector is simply an ordered list of numbers which we can think of as a point in space. For instance, the vector \( \mathbf{v} = [ 2 ] \) is simply a point on the number line at position 2, the vector \( \mathbf{u} = \begin{bmatrix} 1 \\ 3 \end{bmatrix} \) is a point in the two dimensional plane at coordinates (1,3) and the vector \( \mathbf{w} = \begin{bmatrix} 4 \\ 0 \\ -1 \end{bmatrix} \) is a point in three dimensional space at coordinates (4,0,-1). But what about higher dimensions? The spaces in which the dimensions \( \geq 4 \) are called the hyperspace. It may be hard/impossible to visualise as our universe is three dimensional (technically four when considering spacetime), but what we can do is to expect the properties of vectors in higher dimensions to also hold true when compared to lower dimensions, and for any geometric concepts such as the notions of distance and angles, we can generalise them using purely algebraic definitions.