Linear algebra formalism: Difference between revisions

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== Exterior algebra ==

== Wedgies and multivals ==

{{Wikipedia|Exterior algebra}}'''Exterior algebra''' is a branch of linear {{w|Algebra over a field|algebra}} which focuses on combining vectors into structures called "multivectors". Such a combination is called the "exterior product" or "wedge product" and is denoted with <math>\wedge</math>.

In many cases, the same things exterior algebra is used for in music theory can be accomplished using normal linear algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations). Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.

=== Wedge product ===

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For a higher-dimensional multivector with elements c<sub>i, j, k...</sub>, the entries at all permutations of the same indices have the same absolute value, and swapping two indices flips the sign.

=== See also ===

* [[Wedgie]]

* [[Hodge dual]]

* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]

* [[Interior product]]

* [[Recoverability]]

* [[User:Mike Battaglia/Exterior Norm Conjecture Table]]

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 {{Todo|complete section|inline=1}}
-== Exterior algebra ==
+== Wedgies and multivals ==
+{{Wikipedia|Exterior algebra}}'''Exterior algebra''' is a branch of linear {{w|Algebra over a field|algebra}} which focuses on combining vectors into structures called "multivectors". Such a combination is called the "exterior product" or "wedge product" and is denoted with <math>\wedge</math>.
+In many cases, the same things exterior algebra is used for in music theory can be accomplished using normal linear algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations). Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.
 === Wedge product ===
@@ Line 146: / Line 150: @@
 For a higher-dimensional multivector with elements c<sub>i, j, k...</sub>, the entries at all permutations of the same indices have the same absolute value, and swapping two indices flips the sign.
+=== See also ===
+* [[Wedgie]]
+* [[Hodge dual]]
+* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]
+* [[Interior product]]
+* [[Recoverability]]
+* [[User:Mike Battaglia/Exterior Norm Conjecture Table]]<!-- main article -->