A lot of scientific computing eventually boils down to accessing things in structures that look like vectors or matrices. In this module, we will examine the basic syntax to create, interact with, and transpose these structures. This is one of the most foundational module in the class, as we will be using an absurd quantity of vectors and matrices moving forward.

We can start with a simple vector, i.e. a series of numbers (for example) organized in a structure:

[1, 2, 3]

3-element Vector{Int64}:
 1
 2
 3

The way in which this Vector{Int64} is presented to you hides two extremely valuable pieces of information. First, it is a Vector, and the type of the elements in this vector are Int64. Paying attention to the type of things will rapidly become second nature, because it really opens up the full capacities of Julia as an expressive language for research.

The second, and possibly more important piece of information, is that this is a column vector. In other words, when we type [1, 2, 3], Julia understands that we really mean $[1\quad 2\quad 3]^\intercal$. Vectors are represented as columns, the way things should be.

We can of course transpose this vector to make it into a row vector:

[1, 2, 3]'

1×3 adjoint(::Vector{Int64}) with eltype Int64:
 1  2  3

Yes! If you want to transpose a vector, you can add ' at the end of it (or use the transpose function), which is a really nice bit of notation.

But why do we care that the vectors are by default laid out as columns? There are two reasons. First, Julia is “column-major”, meaning that things are stored alongside columns. This will be important to keep in mind when dealing with iteration a little later one (but you can forget it for now).

Second, when it will be the time to multiply things together, it will be important to know if our vectors are represented as columns or rows. For example, the $\mathbf{u}\mathbf{v}’$ operation can be written as:

𝐮 = [1, 2, 3]
𝐯 = [2, 1, 0, 1]

𝐮 * 𝐯'

3×4 Matrix{Int64}:
 2  1  0  1
 4  2  0  2
 6  3  0  3

This is nice! We have a code notation that looks like the way we would write this on paper.

But wait! This operation created a new beast we hadn’t seen yet: a Matrix{Int64}. A Matrix and a Vector are actually very close cousins: both are examples of Arrays. An array with a single dimension is a column vector, an array with two dimensions is a matrix, an array with three dimensions is a cube, etc.

Vector{Int64} == Array{Int64, 1}

true

Matrix{Int64} == Array{Int64, 2}

true

There is no alias for arrays in higher dimensions than 2, but it does not really matter, as we do not need them anyways.

In order to better understand vectors and matrices, let’s get back to our example:

𝐀 = 𝐮 * 𝐯'

3×4 Matrix{Int64}:
 2  1  0  1
 4  2  0  2
 6  3  0  3

We can have a look at the size of these structures:

size(𝐮)

(3,)

In the case of 𝐮, the size is (3,), which means that it has a single dimension, with three elements in it. If we apply the same function to 𝐀, we get:

size(𝐀)

(3, 4)

This matrix has, indeed, three rows and four columns. We can query the number of elements alongside a dimension directly:

size(𝐀, 1)

3

This gives the number of rows (using size(𝐀, 2) would give the number of columns, etc.).

The size (or shape) of an array is something we can manipulate. Let’s say that we have a vector:

𝐰 = [true, false, false, true]

4-element Vector{Bool}:
 1
 0
 0
 1

We can reshape it into an identity matrix, using

reshape(𝐰, (2, 2))

2×2 Matrix{Bool}:
 1  0
 0  1

Note that much like we can transpose vectors, we can transpose matrices:

𝐀'

4×3 adjoint(::Matrix{Int64}) with eltype Int64:
 2  4  6
 1  2  3
 0  0  0
 1  2  3

But pay attention to the type of 𝐀' compared to the following:

transpose(𝐀)

4×3 transpose(::Matrix{Int64}) with eltype Int64:
 2  4  6
 1  2  3
 0  0  0
 1  2  3

By looking into the Julia documentation, it shall be revealed that these operations are meant for linear algebra use only. If you want to pivot a matrix, there is a function to do that:

permutedims(𝐀)

4×3 Matrix{Int64}:
 2  4  6
 1  2  3
 0  0  0
 1  2  3

These three operations give you the same numbers in the same order, but they have different types, because Julia is so very particular about its type system (for good reasons). Note that if you really want to use the ' operator because it looks nice, but you really want a matrix, you can:

Matrix(𝐀')

4×3 Matrix{Int64}:
 2  4  6
 1  2  3
 0  0  0
 1  2  3

In the next module, we will see how we can get values out of arrays, use different indexing systems to access values, and how we can take slices of these objects.