Geometric Algebra Vector Space

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Tutorial by Nicholas Gorski GMan


This tutorial will discuss Vector Space. Vector space is to Geometric Algebra as the x-y plane is to Euclidean coordinates...kind of :)


Contents

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Vector Space

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Basis of vectors

Think about your common point. (3,4) for example. We know this is really just a vector from the origin \left \langle 0,0 \right \rangle to the point \left \langle 3,4 \right \rangle. Vectors are made up of compenents: i, j, k, etc. The vector \left \langle 3,4 \right \rangle can also be written as 3i + 4j. Here, i is a component vector (not to be mistaken with the imaginary number i). That is, i is the vector \left \langle 1,0 \right \rangle, as j is another component vector: \left \langle 0,1 \right \rangle. You can see why this works:

2i + 3j =

2 \left \langle 1,0 \right \rangle + 3 \left \langle 0,1 \right \rangle=

\left \langle 2,0 \right \rangle + \left \langle 0,3 \right \rangle=

\left \langle 2,3 \right \rangle

For a third dimension, the letter k is used. So, vector's are scalars combined with components orientations. They are oriented magnitudes.

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Geometric Algebra Components

In GA, the basis componet's are broken down as:


e1

e2

e3

e4


And so on. e is not e (2.71828183...), simply the letter used. These are just like the components x, y, or z when plotting a graph, or the components i, j, k when dealing with vectors. They are the basic components of Geometric Algebra. So in Geometric Algebra, the 5D vector \left \langle 2,-3,7,10,-2.5 \right \rangle is written as 2e1 - 3e2 + 7e3 + 10e4 - 2.5e5.

Don't let the new notation boggle your mind. These are still just normal variables, and can be added and subtracted like normal. Just like 2x - 3y + 3x + 10y = 5x + 7y, 2e1 + 2.5e2 - 4e1 + 2.5e2 = - 2e1 + 5e2.

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Notation

In GA, notation is as follows:

Scalars- Represented using lower-case Greek. \alpha\ \beta\ \gamma\ \delta\ \epsilon will all be scalars.

Vectors- Represented as a lower-case bold letter. \mathbf{a} \ \mathbf{b} \ \mathbf{c} are all vectors.

Multivectors - Denoted as upper-case bold letters. You will learn about these next. \mathbf{A} \ \mathbf{B} \ \mathbf{C} are all multivectors.

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Moving Forward

In this tutorial you learned what represents the basis components in GA. They are equivilent to x, y, or z in a classical plane. Just as (2,5) maps to (x,y), in GA (2,5) maps to (e1,e2).

Now move onto the: <Outer Product>.

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