# Deep (learning) like Jacques Cousteau - Part 3 - Vectors

(TL;DR: Vectors are ordered lists of numbers.)

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Sector, vector, the lyric inspector

X-Ray vision, power perfectorKool Keith from ‘MC Champion’ by Ultramagnetic MC’s

Last
time,
we learnt about **scalars**. We’ll now start learning about
**vectors!**

# Today’s topic: Vectors

## What’s a Vector?

A vector is essentially **a list of numbers!** They’re normally
depicted as **columns of numbers (i.e. as column vectors)**:

The **order in which each number appears** in our vector is important.
For example:

However:

\[\begin{bmatrix} 1 \\ 2 \\ 3 \\ \end{bmatrix} \neq \begin{bmatrix} 3 \\ 2 \\ 1 \\ \end{bmatrix}\]We can **transpose** vectors so that the **rows become columns**, or so that
**the columns become rows**. We’ll use an upper case “T” to denote the
**transpose operator**:

We now have ourselves a **row vector!** Applying the transpose operation
on our row vector brings us right back to where we started:

## A tiny bit of notation

We will use the notation from *Goodfellow, Ian, et
al.* and refer to vectors using
lower case, bold letters:

The **individual elements** of our vectors are often called
**elements, components or entries**. We’ll follow *Goodfellow, Ian,
et al.* and call them
**elements**. We’ll refer to the \(\boldsymbol{i{\textrm{th}}}\)
element of our vector \(\boldsymbol{x}\) using an **italicised** \(x\) with
a **subscript** indicating the element number:

For example, we’ll represent the second element of our vector \(\boldsymbol{x}\) like this:

\[x_{2}\]## How can we represent vectors in R?

### Common vector types

We have many vector types in R. The vector types that I most commonly use are these:

- numeric,
- character, and
- logical

### Coercing them vectors

In R, **all elements of our vectors must be of the same type.** If a
single element in our vectors violates this condition, the entire vector
is coerced into a **more generic class**. For example, this results in
an **integer vector**:

```
x <- c(2L, 3L, 5L)
class(x)
```

```
## [1] "integer"
```

However, this results in a **numeric vector**:

```
x <- c(2L, 3L, 5L, 7)
class(x)
```

```
## [1] "numeric"
```

This makes sense as we learnt in part one that our set of integers, \(\mathbb{Z}\), is a subset of our set of real numbers, \(\mathbb{R}\). That is:

\[\mathbb{Z} \subset \mathbb{R}\]The \(\subset\) symbol means **“is a proper/strict subset of”**. It
indicates that all elements of the set on the left are contained within
the set on the right.

This means that we can represent both our integers and real numbers in a single vector of type `numeric`

. However, we can’t represent all of our real numbers in a vector type of `integer`

.

### How can we create vectors?

We can create vectors by using the `c()`

function. For example, here is
a **numeric vector**:

```
x <- c(1, 2, 3)
print(x)
```

```
## [1] 1 2 3
```

Notice that when we print it out, it looks like a **row vector**.
However, when we use our vector in a `data.frame`

, we get this:

```
x_df <- data.frame(x)
print(x_df)
```

```
## x
## 1 1
## 2 2
## 3 3
```

Looks like a column to me! *(Note: The numbers in the column on the left are row numbers)*

If we then transpose it, we get this:

```
x_df_transposed <- t(x_df)
print(x_df_transposed)
```

```
## [,1] [,2] [,3]
## x 1 2 3
```

Looks like a row vector now! Exciting! If we transpose it again, we’re back to a column vector:

```
t(x_df_transposed)
```

```
## x
## [1,] 1
## [2,] 2
## [3,] 3
```

# Conclusion

Vectors are ordered lists of numbers. We transposed the hell out of one.

Next time, we’ll **combine** our knowledge of scalars with our new knowledge
of vectors!

Justin