Vectors

I highly reccomend you watch this video before working with Vectors https://www.youtube.com/watch?v=fNk_zzaMoSs

What is a vector?

A vector is a container where order matters and repititions are allowed. An N-vector has n components (elements), each component called
$U_1, U_2, ... , U_n$
Different values in a vector can be the same, so:
$% %]]>$
is entirely possible with Vectors.

A 2-vector represents 2 dimensional space and a 3-vector represents 3-dimensional space.

Doing maths with vectors

Two vectors can be added like so:

The resultant is a new vector, $\vec{W}$.

You can only add two vectors if the vectors have the same number of components.

Scalar multiplcation is where you increase every item in a vector by R. Let R be a real number then:

Vector length can be defined using cardinality. The length of a vector, A, is |A|. The length of a vector has uses this formula:

If you try to calculate the length of a negative vector, you get the length of the positive vector.

Another thing to note is:

To reverse a vector you times each component by -1, making everything negative.

And

Vector Space

Start with some set of numbers such as real numbers, integers, whole numbers, rationals etc.

A set, V, of n-vectors is called a vector space if and only if:

1. Given any number, R, from the original space (real numbers, integers, etc) and $\vec{u} \in V$ then $\vec{u} * R \in V$

The superscript number ontop of a set of numbers such as $\R^n$ is the set of all real valued (numbers that are in the real numbers) n-vectors where each component is in the set of Real numbers.

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