# Linear Algebra - 1

##### Written in Math, Algebra

# Vectors

A vector is an abstraction over some grouping of numbers (or other data perhaps).

Gemotry and graphs are a convenient and human intuitive way of representing vectors, though geometry is not the only way of representing/interpreting vectors, neither is it the source of vectors.

One possible interpretation of this abstraction is to think of vectors as an abstraction of a direction and magnitude. For example **5mph north east (45 °).**

This interpretation works especially well in 2d geometric space and basic physics.

In geometric space, vectors are commonly drawn in relation to the point (0,0). This is referred to as **Standard position**.

# Basic Syntax

Vectors a typically represented as a lower case letter with an arrow above it.

* R^{2}*: Set of all 2-dimensional vectors in real space (real numbers)

* R^{n}*: Set of all n-dimensional vectors in real space (real numbers)

**v ∈ R ^{3}**: Vector v is a vector in 3-dimensional real space (pretend that v has an arrow over it)

# Addition and Scaling

Vector addition is simple - just add the corresponding elements.

`(3,5) + (1,2) = ) (4,7)`

When multiplying a vector by a single value, the value is called a *scalar* because it is *scaling* the vector. The vector will maintain its direction (slope) but go further out.

`(2,4) * 2 = (4,8)`

# Unit Vector

A unit vector is a vector with a magnitude of 1. Unit vectors can be used to describe other vectors.

*Every vector is simply a unit vector scaled up.*

# Basic Parameterization

`S = { c * v | c ∈ R}`

A set `S,of `

every possible vector obtained by multiplying an initial vector `v`

by some a constant `c`

, where `c`

is in the set of real numbers.

When returned, such a set will produce a line.