Vector

Definition

A vector is an ordered collection of scalar values.

Example:

[3, 7, 2]

This vector has three values:

3
7
2

The order matters.

[3, 7, 2] is not the same as [2, 7, 3]

Vector = ordered values treated as one object.

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Why Vectors Exist

A single scalar can describe one quantity.

But many situations require multiple quantities together.

Example:

person:
height = 70
weight = 180
age = 30

As a vector:

[70, 180, 30]

The vector lets the system treat related values as one structured item.

Vectors package multiple measurements into one object.

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Vector As Data

A vector can represent data when each position has a consistent meaning.

Example:

[70, 180, 30]

could mean:

position 1 = height
position 2 = weight
position 3 = age

Without that consistent interpretation, the vector is only a list of numbers.

A vector becomes useful data when its positions are consistently interpreted.

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Vector Dimension

The number of values in a vector is its dimension.

Examples:

[4, 9] has dimension 2
[4, 9, 1] has dimension 3
[4, 9, 1, 8] has dimension 4

Dimension tells us how many components the vector has.

Dimension = number of components.

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Vector vs List

A list is a general ordered collection.

A vector is an ordered collection used as a mathematical object.

Example:

List: ["cat", 5, true]
Vector: [2, 4, 6]

In this chapter, vectors are numerical.

That matters because we can add them, subtract them, scale them, and compare them.

A vector is a list-like structure with numerical operations.

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Vector As State

A state can contain multiple values.

Example:

state = [x, y]
state = [3, 5]

A transition rule can update the vector:

[3, 5] -> [4, 5]

Here the system moved from one vector state to another.

A vector can describe the state of a system with multiple quantities.

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Why Vectors Matter For Later Concepts

Vectors let us move from single values to structured numerical data.

They make it possible to represent:

• positions
• measurements
• features
• states with multiple parts

The important shift is simple:

one number -> many coordinated numbers

Vectors are the first step toward treating data as structured numerical objects.

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Final Summary

• A vector is an ordered collection of scalars.
• Order matters.
• Dimension is the number of components.
• A vector can represent data when each position has a consistent meaning.
• A vector can describe a multi-part state.
• Vectors let computation operate on many related values together.