Thiago P.

Linear Algebra - 1

Written in Math, Algebra

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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.

R2: Set of all 2-dimensional vectors in real space (real numbers)

Rn: Set of all n-dimensional vectors in real space (real numbers)

v ∈ R3: 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.