What Is a Vector?

A vector is an ordered list of numbers with a job.

The numbers are called coordinates. The job tells us what the list means. The same list can describe a point, a direction, a measurement, or a learned representation inside a model.

[2, 4]

By itself, this is only a list. In this chapter, we usually read it as two coordinates:

first coordinate  = 2
second coordinate = 4

That means we can draw it from the origin to the point (2, 4).

a = [2, 4] and b = [4, 1]Drag either endpoint. Drag the plane to pan.

The diagram shows two vectors. Drag an endpoint and watch the coordinates change. The useful habit is to keep the list of numbers connected to the object it represents.

Coordinates

In two dimensions, a vector has two coordinates:

v = [v_1, v_2]

The first coordinate moves along the horizontal axis. The second coordinate moves along the vertical axis.

For example:

v = [2, 4]

means:

move 2 units horizontally
move 4 units vertically

MATH-C02-T02-001Exercise: Read the coordinates

If v = [3, 5], what is the second coordinate?

Your answer

Compute it first, then check your number.

HintRead from left to right

The first coordinate is 3. The second coordinate is the next number.

SolutionCoordinate order

In v = [3, 5], the coordinates are ordered. The first coordinate is 3. The second coordinate is 5.

Order Matters

The vector:

[2, 4]

is not the same as:

[4, 2]

The same two numbers appear, but their jobs have changed. If the first coordinate means horizontal movement, then [2, 4] moves 2 horizontally and 4 vertically. The vector [4, 2] does the opposite.

MATH-C02-T02-002Exercise: Check order

Enter 1 if [2, 4] and [4, 2] are different vectors.

Your answer

Compute it first, then check your number.

HintUse Chapter 1

Vectors behave like ordered collections. Position matters.

SolutionDifferent coordinates

Enter 1. In [2, 4], the first coordinate is 2. In [4, 2], the first coordinate is 4.

The two lists contain the same numbers, but the coordinate positions have changed. For vectors, order is part of the meaning.

Why Vectors Matter

Machine learning turns many things into vectors.

An image patch can become a vector of pixel values.
A token, sentence, or document can become an embedding vector.
A model layer can produce a vector of activations.
A user or item can be represented by a vector of learned features.

Once something is a vector, we can compute with it. We can add it, scale it, measure its length, compare it with another vector, and feed it into a model.

MATH-C02-T02-003Exercise: Attach meaning

A vector is written as [7, 2]. Enter 1 if you still need to know what the two coordinates mean before interpreting it in a real problem.

Your answer

Compute it first, then check your number.

HintAsk what is measured

Are the coordinates pixels, rooms, activations, positions, or something else?

SolutionMeaning attached

Enter 1. The list [7, 2] gives the numbers, but the problem tells us what those numbers mean.

Vector Versus Plain List

A vector is stored like a list, but it is not interpreted like a random list. The coordinates share one coordinate system.

For example:

[temperature, humidity]

and:

[height, weight]

may both have two numbers, but they are not the same kind of vector. Adding them would mix unrelated coordinate meanings.

MATH-C02-T02-005Exercise: Check coordinate meaning

Enter 1 if [20, 70] as [temperature, humidity] should not be added directly to [20, 70] as [height, weight].

Your answer

Compute it first, then check your number.

HintAsk what each position means

The first coordinate in one vector is temperature. The first coordinate in the other vector is height.

SolutionSame shape, different meaning

Enter 1. Both lists have two entries, but their coordinate systems are different. Vector operations should combine coordinates with matching meaning.

Code Mirror

In NumPy, a vector is usually stored as a one-dimensional array:

import numpy as np

v = np.array([2, 4])
print(v)

The output is:

[2 4]

The code is simple because the idea is simple: a vector is a structured list of numbers.

MATH-C02-T02-004Exercise: Read a vector in code

If v = np.array([2, 4]), how many coordinates does v have?

Your answer

Compute it first, then check your number.

HintCount entries

The array contains 2 and 4.

SolutionTwo coordinates

The array contains two numbers:

[2, 4]

So the vector has 2 coordinates. In this lesson, those two coordinates can be read as horizontal and vertical components.

Next, we read the same coordinates in two ways: as position and as direction.