Contraction distributive property in GA

In summary, the property of Left-Contraction for Blades in Geometric Algebra states that it has a distributive property with respect to addition, meaning that the left contraction of the sum of two blades is equal to the sum of the individual left contractions. However, the authors do not provide a proof or any guidance on how to derive this property. One possible explanation is that the left contraction is constructed axiomatically using the bilinear operations of wedge product and scalar product, which results in the left contraction being bilinear as well.
  • #1
mnb96
715
5
Hello,
according to my book of 'Geometric Algebra' the operation of Left-Contraction for Blades has a distributive property in respect to addition. However the authors do not prove it, nor they give the smallest hint on how to derive it.

The property says that:

[tex](\textbf{A+B})|\textbf{C}=\textbf{A}|\textbf{C}+\textbf{B}|\textbf{C}[/tex]

where the symbol | denotes Left-Contraction.
Does anyone have a clue on how to prove that identity?
 
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  • #2
Apparently the answer should be that the left contraction is constructed axiomatically by using the operations [tex]\wedge[/tex] (wedge product) and [tex]\ast[/tex] (scalar product), which are both bilinear. It follows, that the left-contraction must be bilinear too.
 

1. What is the contraction distributive property in GA?

The contraction distributive property in GA is a mathematical rule that allows us to simplify expressions involving contractions, which are shortened versions of two words combined with an apostrophe. It states that when a contraction is expanded, the apostrophe can be distributed to all the letters in the words it replaces.

2. How do you apply the contraction distributive property in GA?

To apply the contraction distributive property in GA, we first identify the contraction in the expression. Then, we replace the contraction with the expanded form, distributing the apostrophe to all the letters in the words it replaces. Finally, we simplify the expression by combining like terms if necessary.

3. Why is the contraction distributive property important in GA?

The contraction distributive property is important in GA because it allows us to simplify expressions involving contractions, making them easier to work with. It also helps us to understand the relationship between contractions and their expanded forms, which can be useful in other areas of mathematics and language.

4. Can the contraction distributive property be applied to all contractions in GA?

Yes, the contraction distributive property can be applied to all contractions in GA. This includes common contractions such as "can't", "won't", and "it's", as well as less common ones like "shouldn't" and "wouldn't". As long as the word being replaced by the apostrophe has two or more letters, the property can be applied.

5. Are there any common mistakes when using the contraction distributive property in GA?

One common mistake when using the contraction distributive property in GA is forgetting to distribute the apostrophe to all the letters in the words it replaces. Another mistake is incorrectly expanding the contraction, such as forgetting to add the apostrophe or expanding it to the wrong words. It's important to be careful and double-check our work when applying this property to avoid these mistakes.

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