Definition

Vectors

Definition

There are orderedthree arraysperspectives ofto singleconsider:

numbers

Physics andperspective

are an example of 1st-order tensor.

Vectors are membersArrows of objects known as vector spaces. A vector space can be thought of as the entire collection of all possible vectors of a particular length (or dimension). The three-dimensional real-valued vector space, denoted by �3 is often used to represent our real-world notion of three-dimensional space mathematically.

More formally a vector space is an �-dimensional Cartesian product of a set with itself, along with proper definitions on how to add vectors and multiply them with scalar values. If all of the scalarspointing in a vector are real-valued then the notation �∈�� states that the (boldface lowercase) vector value � is a member of the �-dimensional vector space of real numbers, ��.

Sometimes it is necessary to identify the components of a vector explicitly. The �th scalar element of a vector is written as ��. Notice that this is non-bold lowercase since the element is a scalar. An �-dimensional vector itself can be explicitly written using the following notation:

�=[�1�2⋮��]

Given that scalars exist to represent values why are vectors necessary? One of the primary use cases for vectors is to represent physical quantities that have both a magnitude and a direction. Scalars are only capable of representing magnitudes.

For instance scalars and vectors encode the difference between the speed of a car and its velocity. The velocity contains not only its speed but also its direction of travel.space. It is notdefined difficultby to

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Vectors gravitationalare an ordered list of numbers 

 

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The Vector will be rooted at the origin, center of space and electromagneticroot forcesof orall windvectors. velocity.Then add the numbers 

2D

 

In machine

learning vectors often represent feature vectors, with their individual components specifying how important a particular feature is. Such features could include relative importance of words in a text document, the intensity of a set of pixels in a two-dimensional image or historical price values for a cross-section of financial instruments.

3D