Basis: Difference between revisions
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{{Wikipedia| Basis (linear algebra) }} | |||
A '''basis''' is a list of vectors that represents the infinite set of vectors that are combinations of them. The corresponding infinite set is called its '''subspace'''. | A '''basis''' is a list of vectors that represents the infinite set of vectors that are combinations of them. The corresponding infinite set is called its '''subspace'''. | ||
The plural of | The plural of ''basis'' is ''bases'' (pronounced BAY-sees, or /ˈbeɪ siz/). | ||
Bases are mathematical structures that come from the field of [[Wikipedia:Linear algebra|linear algebra]], and are used in [[regular temperament theory]], where the most common example of a basis is a [[comma basis]]. The fact that a comma basis is a ''basis'' conveys how when a temperament [[tempers out]] the set of commas explicitly listed in a comma basis, then it also tempers out any interval that's equal to any combination of those commas. We could never possibly list the infinitude of commas tempered out, so instead we carefully choose a minimal set of commas that is capable of representing all of them. | Bases are mathematical structures that come from the field of [[Wikipedia:Linear algebra|linear algebra]], and are used in [[regular temperament theory]], where the most common example of a basis is a [[comma basis]]. The fact that a comma basis is a ''basis'' conveys how when a temperament [[tempers out]] the set of commas explicitly listed in a comma basis, then it also tempers out any interval that's equal to any combination of those commas. We could never possibly list the infinitude of commas tempered out, so instead we carefully choose a minimal set of commas that is capable of representing all of them. | ||
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== Mathematical details == | == Mathematical details == | ||
In mathematical language, a | In mathematical language, a basis for a [[Wikipedia:Linear_subspace|subspace]] of a [[Wikipedia:Vector_space|vector space]] is a minimal set of [[Wikipedia:Vector_(mathematics_and_physics)|vectors]] that [[Wikipedia:Linear_span|span]] the subspace. | ||
For example, a mathematical word for the set of all commas tempered out by a temperament is a "null-space", and specifically this is the null-space of its [[mapping]] matrix; "null-space" uses the word "space" in this same sense of a "subspace". | For example, a mathematical word for the set of all commas tempered out by a temperament is a "null-space", and specifically this is the null-space of its [[mapping]] matrix; "null-space" uses the word "space" in this same sense of a "subspace". | ||