A free resource!

url: https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)
title: Linear Algebra
authors: 
  - David Cherney
  - Tom Denton
  - Andre Waldron

Linear Algebra

1. What is Linear Algebra?

Basically, Linear Algebra is the study of vectors and linear transformations. Vectors typically represent both a magnitude and a direction. Linear Transformations are functions of vectors that respect vector addition.

1.1 What are Vectors?

Vectors are things you can add and scalar multiply.

1.2 What are Linear Functions?

Typical thought of functions is that you push independent variables into a function and it returns a value. In linear algebra, functions take in vectors of some type, and return vectors of the same or possibly another type.

To describe the essential characteristics of linear functions, let $L$ represent a linear function, whose parameters, both inputs and outputs, are vectors abiding by addition and scalar multiplication. Let $u$ and $v$ be vectors, and let $c$ be a number. We know that $u+v$ and $cu$ are also vectors.

Since $L$ is a function of vectors, $L(u)$ is also a vector. Same is true for $L(v)$ , $L(u+v)$ , and $L(cu)$ . We can continue to expand the logic as $L(u) + L(v)$ is a vector, and $cL(u)$ is also a vector.

Let’s put some definitions onto our operations:

Additivity:

L(u+v) = L(u) + L(v)

Homogeneity:

L(cu) = cL(u)

Many functions of vectors do not obey this requirement. However, linear algebra is the study of those that do. Additivity means it doesn’t matter if you add the inputs before passing into the function first, or adding the outputs of the function.

Functions of vectors that obey the additivity and homogeneity properties are linear, hence linear algebra. Additivity and homogeneity together are called linearity. And other names include:

Linear Operator
Homomorphism
Linear Map
Linear Transformation

A really cool example is actually the derivative operator!

\begin{align*} \frac{d}{dx}(cf) &= c\frac{d}{dx}(f) \\ \frac{d}{dx}(f+g) &= \frac{d}{dx}(f) + \frac{d}{dx}(g) \end{align*}

Worth noting, to avoid confusion, for linear maps $L$ , you will see it written as $Lu$ instead of $L(u)$ . This is because the linearity property of a linear transformation $L$ means that $L(u)$ can be thought of as multiplying the vector $u$ by the linear operator $L$ . Another fun example:

L(cu + dv) = cLu + dLv

Note that $uL$ wouldn’t make any sense. However, when written, it almost looks like we are multiplying through my $L$ .

1.3 What is a Matrix?

It is important to note the following:

A = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} \in \mathbb{R}^{2 \times 3}

This is just to visualize the $n \times m$ notation. To me, it’s opposite of the $(x,y)$ you would think graphically. But consider, suppose we have matrix $a \times b$ , if the first number $a$ is for $x$ axis so to speak, then the number represents how many rows of $x$ there are, not necessarily how far along the $x$ -axis to traverse. And if the second is number $b$ is for $y$ , then the same applies. It’s more like a count of the number of $y$ -axes, or columns, we are dealing with.

Just remember that it is not the same as coordinates.

When thinking in terms of identifying an element in a table, it might just be easier to remember it as opposite. And maybe that makes since because then the $y$ value goes downwards as the value increases.