One of the first models learned in physics are the equations governing the laws of motion with constant acceleration: \(x(t) = x_0 + v_0 t + 1/2 \cdot a t^2\). This is a consequence of Newton’s second law of motion applied to the constant acceleration case. A related formula for the velocity is \(v(t) = v_0 + at\). The following figure is produced using these formulas applied to both the vertical position and the horizontal position:
Position, velocity, and acceleration vectors (scaled) for projectile motion. Vectors are drawn with tail on the projectile. The position vector (black) points from the origin to the projectile, the velocity vector (red) is in the direction of the trajectory, and the acceleration vector (green) is a constant pointing downward.
For the motion in the above figure, the object’s \(x\) and \(y\) values change according to the same rule, but, as the acceleration is different in each direction, we get different formula, namely: \(x(t) = x_0 + v_{0x} t\) and \(y(t) = y_0 + v_{0y}t - 1/2 \cdot gt^2\).
It is common to work with both formulas at once. Mathematically, when graphing, we naturally pair off two values using Cartesian coordinates (e.g., \((x,y)\)). Another means of combining related values is to use a vector. The notation for a vector varies, but to distinguish them from a point we will use \(\langle x,~ y\rangle\). With this notation, we can use it to represent the position, the velocity, and the acceleration at time \(t\) through:
\[ \begin{align*} \vec{x} &= \langle x_0 + v_{0x}t,~ -(1/2) g t^2 + v_{0y}t + y_0 \rangle,\\ \vec{v} &= \langle v_{0x},~ -gt + v_{0y} \rangle, \text{ and }\\ \vec{a} &= \langle 0,~ -g \rangle. \end{align*} \]
Don’t spend time thinking about the formulas if they are unfamiliar. The point emphasized here is that we have used the notation \(\langle x,~ y \rangle\) to collect the two values into a single object, which we indicate through a label on the variable name. These are vectors, and we shall see they find use far beyond this application.
Initially, our primary use of vectors will be as containers, but it is worthwhile to spend some time to discuss properties of vectors and their visualization.
A line segment in the plane connects two points \((x_0, y_0)\) and \((x_1, y_1)\). The length of a line segment (its magnitude) is given by the distance formula \(\sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}\). A line segment can be given a direction by assigning an initial point and a terminal point. A directed line segment has both a direction and a magnitude. A vector is an abstraction where just these two properties \(-\) a direction and a magnitude \(-\) are intrinsic. While a directed line segment can be represented by a vector, a single vector describes all such line segments found by translation. That is, how the the vector is located when visualized is for convenience, it is not a characteristic of the vector. In the figure above, all vectors are drawn with their tails at the position of the projectile over time.
We can visualize a (two-dimensional) vector as an arrow in space. This arrow has two components. We represent a vector mathematically as \(\langle x,~ y \rangle\). For example, the vector connecting the point \((x_0, y_0)\) to \((x_1, y_1)\) is \(\langle x_1 - x_0,~ y_1 - y_0 \rangle\).
The magnitude of a vector comes from the distance formula applied to a line segment, and is \(\| \vec{v} \| = \sqrt{x^2 + y^2}\).
A vector and its unit vector. They share the same direction, but the unit vector has a standardized magnitude.
We call the values \(x\) and \(y\) of the vector \(\vec{v} = \langle x,~ y \rangle\) the components of the \(v\).
Two operations on vectors are fundamental.
The sum of two vectors can be visualized by placing the tail of one at the tip of the other
The difference of two vectors can be visualized by placing the tail of one at the tip of the other
The concept of scalar multiplication and addition, allow the decomposition of vectors into standard vectors. The standard unit vectors in two dimensions are \(e_x = \langle 1,~ 0 \rangle\) and \(e_y = \langle 0,~ 1 \rangle\). Any two dimensional vector can be written uniquely as \(a e_x + b e_y\) for some pair of scalars \(a\) and \(b\) (or as, \(\langle a, b \rangle\)). This is true more generally where the two vectors are not the standard unit vectors - they can be any two non-parallel vectors.
The vector \(\langle 4,3 \rangle\) is written as \(2/3 \cdot\langle 1,2 \rangle + 5/3 \cdot\langle 2,1 \rangle\). Any vector \(\vec{c}\) can be written uniquely as \(\alpha\cdot\vec{a} + \beta \cdot \vec{b}\) provided \(\vec{a}\) and \(\vec{b}\) are not parallel.
The two operations of scalar multiplication and vector addition are defined in a component-by-component basis. We will see that there are many other circumstances where performing the same action on each component in a vector is desirable.
When a vector is placed with its tail at the origin, it can be described in terms of the angle it makes with the \(x\) axis, \(\theta\), and its length, \(r\). The following formulas apply:
\[ r = \sqrt{x^2 + y^2}, \quad \tan(\theta) = y/x. \]
If we are given \(r\) and \(\theta\), then the vector is \(v = \langle r \cdot \cos(\theta),~ r \cdot \sin(\theta) \rangle\).
A vector \(\langle x, y \rangle\) can be written as \(\langle r\cdot \cos(\theta), r\cdot\sin(\theta) \rangle\) for values \(r\) and \(\theta\). The value \(r\) is a magnitude, the direction parameterized by \(\theta\).
A vector in Julia can be represented by its individual components, but it is more convenient to combine them into a collection using the [,] notation:
x, y = 1, 2
v = [x, y] # square brackets, not angles2-element Vector{Int64}:
1
2The basic vector operations are implemented for vector objects. For example, the vector v has scalar multiplication defined for it:
10 * v2-element Vector{Int64}:
10
20The norm function returns the magnitude of the vector (by default):
import LinearAlgebra: normnorm(v)2.23606797749979A unit vector is then found by scaling by the reciprocal of the magnitude:
v / norm(v)2-element Vector{Float64}:
0.4472135954999579
0.8944271909999159In addition, if w is another vector, we can add and subtract:
w = [3, 2]
v + w, v - 2w([4, 4], [-5, -2])We see above that scalar multiplication, addition, and subtraction can be done without new notation. This is because the usual operators have methods defined for vectors.
Finally, to find an angle \(\theta\) from a vector \(\langle x,~ y\rangle\), we can employ the atan function using two arguments:
norm(v), atan(y, x) # v = [x, y](2.23606797749979, 1.1071487177940904)Mathematically, vectors can be generalized to more than \(2\) dimensions. For example, using \(3\)-dimensional vectors are common when modeling events happening in space, and \(4\)-dimensional vectors are common when modeling space and time.
In Julia there are many uses for vectors outside of physics applications. A vector in Julia is just a one-dimensional collection of similarly typed values and a special case of an array. Such objects find widespread usage. For example:
Julia, vectors are used to hold the \(x\) and \(y\) coordinates of a collection of points to plot and connect with straight lines. There can be hundreds of such points in a plot.Creating higher-dimensional vectors is similar to creating a two-dimensional vector, we just include more components:
fibs = [1, 1, 2, 3, 5, 8, 13]7-element Vector{Int64}:
1
1
2
3
5
8
13Later we will discuss different ways to modify the values of a vector to create new ones, similar to how scalar multiplication does.
As mentioned, vectors in Julia are comprised of elements of a similar type, but the type is not limited to numeric values. Some examples:, * a vector of strings might be useful for text processing, For example, the WordTokenizers.jl package takes text and produces tokens from the words. * a vector of Boolean values can naturally arise and is widely used within Julia’s DataFrames.jl package. * some applications are even naturally represented in terms of vectors of vectors (such as happens when plotting a collection points).
Look at the output of these two vectors, in particular how the underlying type of the components is described on printing.
["one", "two", "three"] # Array{T, 1} is shorthand for Vector{T}. Here T - the type - is String3-element Vector{String}:
"one"
"two"
"three"[true, false, true] # vector of Bool values3-element Vector{Bool}:
1
0
1Finally, we mention that if Julia has values of different types it will promote them to a common type if possible. Here we combine three types of numbers, and see that each is promoted to Float64:
[1, 2.0, 3//1]3-element Vector{Float64}:
1.0
2.0
3.0Whereas, in this example where there is no common type to promote the values to, a catch-all type of Any is used to hold the components.
["one", 2, 3.0, 4//1]4-element Vector{Any}:
"one"
2
3.0
4//1Getting the components out of a vector can be done in a manner similar to multiple assignment:
vs = [1, 2]
v₁, v₂ = vs2-element Vector{Int64}:
1
2When the same number of variable names are on the left hand side of the assignment as in the container on the right, each is assigned in order.
Though this is convenient for small vectors, it is far from being so if the vector has a large number of components. However, the vector is stored in order with a first, second, third, \(\dots\) component. Julia allows these values to be referred to by index. This too uses the [] notation, though differently. Here is how we get the second component of vs:
vs[2]2The last value of a vector is usually denoted by \(v_n\). In Julia, the length function will return \(n\), the number of items in the container. So v[length(v)] will refer to the last component. However, the special keyword end will do so as well, when put into the context of indexing. So v[end] is more idiomatic. (Similarly, there is a begin keyword that is useful when the vector is not \(1\)-based, as is typical but not mandatory.)
There is much more to indexing than just indexing by a single integer value. For example, the following can be used for indexing:
Some add-on packages extend this further.
Indexing notation can also be used with assignment, meaning it can appear on the left hand side of an equals sign. The following expression replaces the second component with a new value:
vs[2] = 1010The value of the right hand side is returned, not the value for vs. We can check that vs is then \(\langle 1,~ 10 \rangle\) by showing it:
vs = [1,2]
vs[2] = 10
vs2-element Vector{Int64}:
1
10The assignment vs[2] is different than the initial assignment vs=[1,2] in that, vs[2]=10 modifies the container that vs points to, whereas vs=[1,2] replaces any binding for vs. The indexed assignment is more memory efficient when vectors are large. This point is also of interest when passing vectors to functions, as a function may modify components of the vector passed to it, though can’t replace the container itself.
As mentioned, the length function returns the number of components in a vector. It is one of several useful functions for vectors.
The sum and prod function will add and multiply the elements in a vector:
v1 = [1,1,2,3,5,8]
sum(v1), prod(v1)(20, 240)The unique function will throw out any duplicates:
unique(v1) # drop a `1`5-element Vector{Int64}:
1
2
3
5
8The functions maximum and minimum will return the largest and smallest values of an appropriate vector.
maximum(v1)8(These should not be confused with max and min which give the largest or smallest value over all their arguments.)
The extrema function returns both the smallest and largest value of a collection:
extrema(v1)(1, 8)Consider now
𝒗 = [1,4,2,3]4-element Vector{Int64}:
1
4
2
3The sort function will rearrange the values in 𝒗:
sort(𝒗)4-element Vector{Int64}:
1
2
3
4The keyword argument, rev=true can be given to get values in decreasing order:
sort(𝒗, rev=true)4-element Vector{Int64}:
4
3
2
1For adding a new element to a vector the push! method can be used, as in
push!(𝒗, 5)5-element Vector{Int64}:
1
4
2
3
5To append more than one value, the append! function can be used:
append!(v1, [6,8,7])9-element Vector{Int64}:
1
1
2
3
5
8
6
8
7These two functions modify or mutate the values stored within the vector 𝒗 that passed as an argument. In the push! example above, the value 5 is added to the vector of \(4\) elements. In Julia, a convention is to name mutating functions with a trailing exclamation mark. (Again, these do not mutate the binding of 𝒗 to the container, but do mutate the contents of the container.) There are functions with mutating and non-mutating definitions, an example is sort and sort!.
If only a mutating function is available, like push!, and this is not desired a copy of the vector can be made. It is not enough to copy by assignment, as with w = 𝒗. As both w and 𝒗 will be bound to the same memory location. Rather, you call copy (or sometimes deepcopy) to make a new container with copied contents, as in w = copy(𝒗).
Creating new vectors of a given size is common for programming, though not much use will be made here. There are many different functions to do so: ones to make a vector of ones, zeros to make a vector of zeros, trues and falses to make Boolean vectors of a given size, and similar to make a similar-sized vector (with no particular values assigned).
Functions such as sum or length are known as reductions as they reduce the “dimensionality” of the data: a vector is in some sense \(1\)-dimensional, the sum or length are \(0\)-dimensional numbers. Applying a reduction is straightforward – it is just a regular function call.
v = [1, 2, 3, 4]
sum(v), length(v)(10, 4)Other desired operations with vectors act differently. Rather than reduce a collection of values using some formula, the goal is to apply some formula to each of the values, returning a modified vector. A simple example might be to square each element, or subtract the average value from each element. An example comes from statistics. When computing a variance, we start with data \(x_1, x_2, \dots, x_n\) and along the way form the values \((x_1-\bar{x})^2, (x_2-\bar{x})^2, \dots, (x_n-\bar{x})^2\).
Such things can be done in many different ways. Here we describe two, but will primarily utilize the first.
If we have a vector, xs, and a function, f, to apply to each value, there is a simple means to achieve this task. By adding a “dot” between the function name and the parenthesis that enclose the arguments, instructs Julia to “broadcast” the function call. The details allow for more flexibility, but, for this purpose, broadcasting will take each value in xs and apply f to it, returning a vector of the same size as xs. When more than one argument is involved, broadcasting will try to fill out different sized objects.
For example, the following will find, using sqrt, the square root of each value in a vector:
xs = [1, 1, 3, 4, 7]
sqrt.(xs)5-element Vector{Float64}:
1.0
1.0
1.7320508075688772
2.0
2.6457513110645907This would find the sine of each number in xs:
sin.(xs)5-element Vector{Float64}:
0.8414709848078965
0.8414709848078965
0.1411200080598672
-0.7568024953079282
0.6569865987187891For each function, the .( (and not () after the name is the surface syntax for broadcasting.
The ^ operator is an infix operator. Infix operators can be broadcast, as well, by using the form . prior to the operator, as in:
xs .^ 25-element Vector{Int64}:
1
1
9
16
49Here is an example involving the logarithm of a set of numbers. In astronomy, a logarithm with base \(100^{1/5}\) is used for star brightness. We can use broadcasting to find this value for several values at once through:
ys = [1/5000, 1/500, 1/50, 1/5, 5, 50]
base = (100)^(1/5)
log.(base, ys)6-element Vector{Float64}:
-9.247425010840049
-6.747425010840047
-4.247425010840047
-1.747425010840047
1.747425010840047
4.247425010840047Broadcasting with multiple arguments allows for mixing of vectors and scalar values, as above, making it convenient when parameters are used.
As a final example, the task from statistics of centering and then squaring can be done with broadcasting. We go a bit further, showing how to compute the sample variance of a data set. This has the formula
\[ \frac{1}{n-1}\cdot ((x_1-\bar{x})^2 + \cdots + (x_n - \bar{x})^2). \]
This can be computed, with broadcasting, through:
import Statistics: mean
xs = [1, 1, 2, 3, 5, 8, 13]
n = length(xs)
(1/(n-1)) * sum(abs2.(xs .- mean(xs)))19.57142857142857This shows many of the manipulations that can be made with vectors. Rather than write .^2, we follow the definition of var and chose the possibly more performant abs2 function which, in general, efficiently finds \(|x|^2\) for various number types. The .- uses broadcasting to subtract a scalar (mean(xs)) from a vector (xs). Without the ., this would error.
The map function is very much related to broadcasting and similarly named functions are found in many different programming languages. (The “dot” broadcast is mostly limited to Julia and mirrors a similar usage of a dot in MATLAB.) For those familiar with other programming languages, using map may seem more natural. Its syntax is map(f, xs).
In mathematics, set notation is often used to describe elements in a set.
For example, the first \(5\) cubed numbers can be described by:
\[ \{x^3: x \text{ in } 1, 2,\dots, 5\} \]
Comprehension notation is similar. The above could be created in Julia with:
xs = [1,2,3,4,5]
[x^3 for x in xs]5-element Vector{Int64}:
1
8
27
64
125Something similar can be done more succinctly:
xs .^ 35-element Vector{Int64}:
1
8
27
64
125However, comprehensions have a value when more complicated expressions are desired as they work with an expression of xs, and not a pre-defined or user-defined function.
Another typical example of set notation might include a condition, such as, the numbers divisible by \(7\) between \(1\) and \(100\). Set notation might be:
\[ \{x: \text{rem}(x, 7) = 0 \text{ for } x \text{ in } 1, 2, \dots, 100\}. \]
This would be read: “the set of \(x\) such that the remainder on division by \(7\) is \(0\) for all x in \(1, 2, \dots, 100\).”
In Julia, a comprehension can include an if clause to mirror, somewhat, the math notation. For example, the above would become (using 1:100 as a means to create the numbers \(1,2,\dots, 100\), as will be described in an upcoming section):
[x for x in 1:100 if rem(x,7) == 0]14-element Vector{Int64}:
7
14
21
28
35
42
49
56
63
70
77
84
91
98Comprehensions can be a convenient means to describe a collection of numbers, especially when no function is defined, but the simplicity of the broadcast notation (just adding a judicious “.”) leads to its more common use in these notes.
The process of plotting a function is usually first taught by generating a “T” table: values of \(x\) and corresponding values of \(y\). These pairs are then plotted on a Cartesian grid and the points are connected with lines to form the graph. Generating a “T” table in Julia is easy: create the \(x\) values, then create the \(y\) values for each \(x\).
To be concrete, let’s generate \(7\) points to plot \(f(x) = x^2\) over \([-1,1]\).
The first task is to create the data. We will soon see more convenient ways to generate patterned data, but for now, we do this by hand:
a, b, n = -1, 1, 7
d = (b-a) // (n-1)
xs = [a, a+d, a+2d, a+3d, a+4d, a+5d, a+6d] # 7 points7-element Vector{Rational{Int64}}:
-1
-2//3
-1//3
0
1//3
2//3
1To get the corresponding \(y\) values, we can use a compression (or define a function and broadcast):
ys = [x^2 for x in xs]7-element Vector{Rational{Int64}}:
1
4//9
1//9
0
1//9
4//9
1Vectors can be compared together by combining them into a separate container, as follows:
[xs ys]7×2 Matrix{Rational{Int64}}:
-1 1
-2//3 4//9
-1//3 1//9
0 0
1//3 1//9
2//3 4//9
1 1(If there is a space between objects they are horizontally combined. In our construction of vectors using [] we used a comma for vertical combination. More generally we should use a ; for vertical concatenation.)
In the sequel, we will typically use broadcasting for this task using two steps: one to define a function the second to broadcast it.
The style generally employed here is to use plural variable names for a collection of values, such as the vector of \(y\) values and singular names when a single value is being referred to, leading to expressions like “x in xs”.
We end this section with some general comments that are for those interested in a bit more, but in general aren’t needed to understand most all of what follows later.
Vectors in Julia are a container for values. Vectors are one of many different types of containers. The Julia manual uses the word “collection” to refer to a container of values that has properties like a vector. Here we briefly review some alternate container types that are common in Julia and find use in these notes.
First, here are some of the properties of a vector:
Vectors are homogeneous. That is, the container holding the vectors all have a common type. This type might be an abstract type, but for high performance, concrete types (like 64-bit floating point or 64-bit integers) are more typical.
Vectors are \(1\)-dimensional.
Vectors are ordered and indexable by their order. In Julia, the default indexing for vectors is \(1\)-based (starting) with one, with numeric access to the first, second, third, …, last entries.
Vectors are mutable. That is, their elements may be changed; the container may be grown or shrunk
Vectors are iterable. That is, their values can be accessed one-by-one in various manners.
These properties may not all be desirable for one reason or the other and Julia has developed a large number of alternative container types of which we describe a few here.
Vectors are \(1\)-dimensional, but there are desires for other dimensions. Vectors are a implemented as a special case of a more general array type. Arrays are of dimension \(N\) for various non-negative values of \(N\). A common, and somewhat familiar, mathematical use of a \(2\)-dimensional array is a matrix.
Arrays can have their entries accessed by dimension and within that dimension their components. By default these are \(1\)-based, but other offsets are possible through the OffetArrays.jl package. A matrix can refer to its values either by row and column indices or, as a matrix has linear indexing by a single index.
For large collections of data with many entries being \(0\) a sparse array is beneficial for less memory intensive storage. These are implemented in the SparseArrays.jl package.
There are numerous array types available. Julia has a number of generic methods for working with different arrays. An example would be eachindex, which provides an iterator interface to the underlying array access by index in an efficient manner.
Tuples are fixed-length containers where there is no expectation or enforcement of their having a common type. Tuples just combine values together in an immutable container. Like vectors they can be accessed by index (also \(1\)-based). Unlike vectors, the containers are immutable - elements can not be changed and the length of the container may not change. This has benefits for performance purposes. (For fixed length, mutable containers that have the benefits of tuples and vectors, the StaticArrays.jl package is available).
While a vector is formed by placing comma-separated values within a [] pair (e.g., [1,2,3]), a tuple is formed by placing comma-separated values within a () pair. A tuple of length \(1\) uses a convention of a trailing comma to distinguish it from a parenthesized expression (e.g. (1,) is a tuple, (1) is just the value 1).
Technically, the tuple is formed just by the use of commas, which separate different expressions. The parentheses are typically used, as they clarify the intent and disambiguate some usage. In a notebook interface, it is useful to just use commas to separate values to output, as typically the only the last command is displayed. This usage just forms a tuple of the values and displays that.
There are named tuples where each component has an associated name. Like a tuple these can be indexed by number and unlike regular tuples also by name.
For example, here a named tuple is constructed, and then its elements referenced:
nt = (one=1, two="two", three=:three) # heterogeneous values (Int, String, Symbol)
nt.one, nt[2], nt[end] # named tuples have name or index access(1, "two", :three)A named tuple is a container that allows access by index or by name. They are easily constructed. For example:
nt = (x0 = 1, x1 = 4, y0 = 2, y1 = 6)(x0 = 1, x1 = 4, y0 = 2, y1 = 6)The values in a named tuple can be accessed using the “dot” notation:
nt.x14Alternatively, the index notation – using a symbol for the name – can be used:
nt[:x1]4Named tuples are employed to pass parameters to functions. To find the slope, we could do:
(nt.y1 - nt.y0) / (nt.x1 - nt.x0)1.3333333333333333However, more commonly used is destructuring, where named variables are extracted by name when the left hand side matches the right hand side:
(;x0, x1) = nt # only extract what is desired
x1 - x03(This works for named tuples and other iterable containers in Julia. It also works the other way, if x0 and x1 are defined then (;x0, x1) creates a named tuple with those values.)
Named tuples associate a name (in this case a symbol) to a value. More generally an associative array associates to each key a value, where the keys and values may be of different types.
The pair notation, key => value, is used to make one association. A dictionary is used to have a container of associations. For example, this constructs a simple dictionary associating a spelled out name with a numeric value:
d = Dict("one" => 1, "two" => 2, "three" => 3)Dict{String, Int64} with 3 entries:
"two" => 2
"one" => 1
"three" => 3The print out shows the keys are of type String, the values of type Int64, in this case. There are a number of different means to construct dictionaries.
The values in a dictionary can be accessed by name:
d["two"]2Named tuples are associative arrays where the keys are restricted to symbols. There are other types of associative arrays, specialized cases of the AbstractDict type with performance benefits for specific use cases. In these notes, dictionaries appear as output in some function calls.
Unlike vectors and tuples, named tuples and dictionaries are not currently supported by broadcasting. This causes no loss in usefulness, as the values can easily be iterated over, but the convenience of the dot notation is lost.
Which command will create the vector \(\vec{v} = \langle 4,~ 3 \rangle\)?
Which command will create the vector with components “4,3,2,1”?
What is the magnitude of the vector \(\vec{v} = \langle 10,~ 15 \rangle\)?
Which of the following is the unit vector in the direction of \(\vec{v} = \langle 3,~ 4 \rangle\)?
What vector is in the same direction as \(\vec{v} = \langle 3,~ 4 \rangle\) but is 10 times as long?
If \(\vec{v} = \langle 3,~ 4 \rangle\) and \(\vec{w} = \langle 1,~ 2 \rangle\) find \(2\vec{v} + 5 \vec{w}\).
Let v be defined by:
v = [1, 1, 2, 3, 5, 8, 13, 21]What is the length of v?
What is the sum of v?
What is the prod of v?
From transum.org.
The figure shows \(5\) vectors.
Express vector c in terms of a and b:
Express vector d in terms of a and b:
Express vector e in terms of a and b:
If xs=[1, 2, 3, 4] and f(x) = x^2 which of these will not produce the vector [1, 4, 9, 16]?
Let \(f(x) = \sin(x)\) and \(g(x) = \cos(x)\). In the interval \([0, 2\pi]\) the zeros of \(g(x)\) are given by
zs = [pi/2, 3pi/2]2-element Vector{Float64}:
1.5707963267948966
4.71238898038469What construct will give the function values of \(f\) at the zeros of \(g\)?
If zs = [1,4,9,16] which of these commands will return [1.0, 2.0, 3.0, 4.0]?