Introduction to Vectors and Matrices

In this section, we will introduce two concepts that are pretty foundational to the study of linear algebra: vectors and matrices. It’s important to recognize that these concepts are fundamentally related; I’d personally recommend thinking about vectors as inputs and output, while matrices are the actions taken on those inputs and outputs.

With this frame of thought in mind, matrices act on vectors, so it makes sense to get started with vectors to understand matrices. This might seem somewhat disconnected from section one and section two, but we’ll bring it all together in the next section.

Vectors

There are lots of different ways to think about vectors. Two common ones are what 3blue1brown refers to as the “computer science student perspective” and the “physics student perspective.” (Small side note, but that link is to a video series that gives a very helpful visual explanation of lots of liner algebra topics! If you’re confused by something in these notes, a watch of the relevant video will likely help.) The computer science student perspective is to think about vectors as lists of numbers; essentially a coordinate pair with potentially more than just two entries. The physics student perspective, however, is to think of vectors as arrows in space, pointing a certain direction with a certain length. This makes most sense with two- and three-dimensional vectors, but works generally, if you’re willing to bend your brain a bit.

Definition

This is the definition that we will be using, and will be most common in an introductory linear algebra course.¹ We’ll refer to vectors using boldface type (e.g., $\mathbf{v}$) to distinguish them from “scalars” (pronounced “scale -er”, which is, in this context,² a synonym for a real number), another common notation is to use a small arrow over the letter, which is easier to write by hand.

In my opinion (as a computer science student), it’s easiest to start with the computer science student perspective; a vector is a list of $n$ real numbers, for some $n>0$. We have two ways to write a given vector $\mathbf{v}$ in this “list of numbers” way: the helpful way and the lazy way. We’ll use the variables $a_1$, $a_2$, …, $a_n$ to be the $n$ numbers that make up our vector (the numbers that make up a vector are called its components). When combining matrices and vectors (i.e., what almost all the rest of these notes are about), the computations are easy to follow if you write your vectors in the helpful way:

$$\mathbf{v}=\left[\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_n\end{array}\right]$$

This is extremely inconvenient, however, when talking about a vector in the middle of a paragraph. This is where we get the lazy way: $\mathbf{v}={[a_1,a_2,\dots ,a_n]}^{\mathsf{T}}$, which is equivalent to the version above. The small “T” means transpose, and has a specific meaning that’s important when it’s applied to matrices, but you can think of it as just flipping something over for now.

It’s common to want to talk about the set (if you’ve not encountered this word in a mathmatics context before, you can just think of the term “collection”) of all vectors with $m$ components. We’ll use the symbol ${\mathbb{R}}^m$ for this purpse, so $\mathbf{v}\in {\mathbb{R}}^m$ means that $\mathbf{v}$ has $m$ components (the symbol $\in$ is read as “in,” so $\mathbf{v}\in {\mathbb{R}}^m$ means that $\mathbf{v}$ is in our collection of vectors with $m$ components).

This is a lot of notation, especially if you haven’t seen this stuff before! It might take some practice to really feel fluent in all of these symbols, but they’re there to help — more practice will make it easier.

Basic Vector Operations

Vectors, on their own, aren’t particularly useful. Thus far, it’s just a list of numbers! The concept of $3$ isn’t very useful in isolation, and like numbers, vectors are only really useful when you’re able to actually do things with them. We’re going to define two operations that involve vectors, scalar multiplication and vector addition. (You may have heard of the dot- and cross-products; we’ll discuss these later.)

Scalar Multiplication

Definition

Given an arbitrary scalar (again, a scalar in this context is just some real number) that we’ll call $c$ and some vector $\mathbf{v}\in {\mathbb{R}}^n$ (remember, this means a “vector called $\mathbf{v}$ in ${\mathbb{R}}^n$, the collection/set of vectors that have $n$ components”), we define the scalar product of $c$ and $\mathbf{v}$, denoted $c\mathbf{v}$ as:

$$c\mathbf{v}=\left[\begin{array}{c}c{a}_1\\ c{a}_2\\ ⋮\\ c{a}_n\end{array}\right]$$

where $a_i$ are the compenets of $\mathbf{v}$.

So, what does this kludge of notation really mean? Well, $\mathbf{v}$ is a vector of length $n$, and it has components $a_1$, $a_2$, …, $a_n$. When we multiply $\mathbf{v}$ by $c$, we get another vector, also of length $n$, but each of the compoenets of our new vector have been multiplied by $c$. So, in other words, the scalar-vector product just multiplies each component of a given vector by some scalar.

Example

Let’s have our scalar be $3$ and our vector $\mathbf{v}={[1,2,3]}^{\mathsf{T}}$. Then we have:

$$\begin{array}{rl}3\mathbf{v}& =3\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\\ & =\left[\begin{array}{c}3\times 1\\ 3\times 2\\ 3\times 3\end{array}\right]\\ & =\left[\begin{array}{c}3\\ 6\\ 9\end{array}\right]\end{array}$$

Practice

Compute:

1. $3\times {[3,2,1]}^{\mathsf{T}}$, remember the left-most component of a vector written in the “lazy” way is the top-most component of a vector written in the helpful way. If you’re hand-writing these, you should try and use the helpful way as much as possible!
2. $\sqrt{7}\times {[\sqrt{7},\sqrt{14},\frac{1}{2}]}^{\mathsf{T}}$, and remember these are just like any other numbers!

Solutions

Vector Addition

Definition

Given two vectors $\mathbf{x}$ and $\mathbf{y}$, both in ${\mathbb{R}}^m$, we define the vector sum of $\mathbf{x}$ and $\mathbf{y}$, denoted $\mathbf{x}+\mathbf{y}$ as:

$$\mathbf{x}+\mathbf{y}=\left[\begin{array}{c}{x}_1+y_1\\ x_2+y_2\\ ⋮\\ x_m+y_m\end{array}\right]$$

where $x_1$, $x_2$, …, $x_m$ are the components of $\mathbf{x}$ and $y_1$, $y_2$, …, $y_m$ are the components of $\mathbf{y}$.

Again, this is a big pile of notation: what’s the important bits? (Notice, I’m not going to just rephrase the definition again like I did earlier, knowing how to read these terse mathematical definitions is a skill, which takes practice to learn like any other.) First, both $\mathbf{x}$ and $\mathbf{y}$ are in ${\mathbb{R}}^m$, so they both have $m$ components (i.e., the same number of components). Next, when we add $\mathbf{x}$ and $\mathbf{y}$, we just add the first components of each to become the first component of our new vector, then the second, etc.

Example

Let’s add $\mathbf{x}={[2,3,7]}^{\mathsf{T}}$ and $\mathbf{y}={[5,4,3]}^{\mathsf{T}}$. Applying our definition we get:

$$\begin{array}{rl}\mathbf{x}+\mathbf{y}& =\left[\begin{array}{c}2\\ 3\\ 7\end{array}\right]+\left[\begin{array}{c}5\\ 4\\ 3\end{array}\right]\\ & =\left[\begin{array}{c}2+5\\ 3+4\\ 7+3\end{array}\right]\\ & =\left[\begin{array}{c}7\\ 7\\ 10\end{array}\right]\end{array}$$

Practice

1. Compute ${[3,1,2]}^{\mathsf{T}}+{[5,3,1]}^{\mathsf{T}}$.
2. We’ve defined vector addition but not vector subtraction. Given both vector addition and scalar multiplication, what would be a sensible way to define vector subtraction? Using this method, compute ${[2,1,3]}^{\mathsf{T}}-{[4,0,1]}^{\mathsf{T}}$.

Solutions

Visualizing Vectors

Intuition

One of the most key insights in linear algebra is that the two perspectives we’ve discussed for thinking about vectors are really the same, and that it’s extremely useful to be able to switch between them. Being able to change your frame of mind at will is a very difficult skill to master, but it’s very worth it (as well as quite necessary for success in a linear algebra course).

With this in mind, let’s consider the “physics student perspective” for conceptualizing vectors, which is much more amenable to visualization.

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