Linear dependence: Difference between revisions

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Linear dependence is defined on sets of basis matrices (matrices acting as bases), such as two temperaments' mappings<ref>Mappings are not typically thought of as bases, but their row vectors can be considered to span rowspaces in an analogous way that comma bases span spaces.</ref>, or two temperaments' comma bases.

A set of basis matrices are linear dependent upon each other when some vector can be found where each basis matrix can produce this vector through a linear combination of its own constituent basis vectors. For a very simple example, the mappings {{~~ket~~|{{map|5 8 12}} {{map|7 11 16}}}} and {{~~ket~~|{{map|7 11 16}} {{map|15 24 35}}}} are linearly dependent because both mappings contain the vector {{map|7 11 16}}. For a less obvious example, the mappings {{~~ket~~|{{map|1 0 -4}} {{map|0 1 4}}}} and {{~~ket~~|{{map|1 2 3}} {{map|0 3 5}}}} are also linearly dependent, because the vector {{map|7 11 16}} can be found through linear combinations of each of their rows; in the first mapping's case, {{map|7 11 16}} = 7{{map|1 0 -4}} + 11{{map|0 1 4}}, and in the second mapping's case, {{map|7 11 16}} = 7{{map|1 2 3}} + -1{{map|0 3 5}}.

A set of basis matrices are linear dependent upon each other when some vector can be found where each basis matrix can produce this vector through a linear combination of its own constituent basis vectors. For a very simple example, the mappings {{rket|{{map|5 8 12}} {{map|7 11 16}}}} and {{rket|{{map|7 11 16}} {{map|15 24 35}}}} are linearly dependent because both mappings contain the vector {{map|7 11 16}}. For a less obvious example, the mappings {{rket|{{map|1 0 -4}} {{map|0 1 4}}}} and {{rket|{{map|1 2 3}} {{map|0 3 5}}}} are also linearly dependent, because the vector {{map|7 11 16}} can be found through linear combinations of each of their rows; in the first mapping's case, {{map|7 11 16}} = 7{{map|1 0 -4}} + 11{{map|0 1 4}}, and in the second mapping's case, {{map|7 11 16}} = 7{{map|1 2 3}} + -1{{map|0 3 5}}.

Sometimes basis matrices can share not just one basis vector, but multiple basis vectors. For example, the comma basis ~~{{bra|~~{{vector|-30 19 0 0}} {{vector|-26 15 1 0}} {{vector|-17 9 0 1}}}} and the comma basis ~~{{bra|~~{{vector|-19 12 0 0}} {{vector|-15 8 1 0}} {{vector|-6 2 0 1}}}} share both the vector {{vector|4 -4 1 0}} as well as the vector {{vector|13 -10 0 1}}:

Sometimes basis matrices can share not just one basis vector, but multiple basis vectors. For example, the comma basis [{{vector|-30 19 0 0}} {{vector|-26 15 1 0}} {{vector|-17 9 0 1}}] and the comma basis [{{vector|-19 12 0 0}} {{vector|-15 8 1 0}} {{vector|-6 2 0 1}}] share both the vector {{vector|4 -4 1 0}} as well as the vector {{vector|13 -10 0 1}}:

* {{vector|4 -4 1 0}} = {{vector|-26 15 1 0}} - {{vector|-30 19 0 0}}

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 Linear dependence is defined on sets of basis matrices (matrices acting as bases), such as two temperaments' mappings<ref>Mappings are not typically thought of as bases, but their row vectors can be considered to span rowspaces in an analogous way that comma bases span spaces.</ref>, or two temperaments' comma bases.
-A set of basis matrices are linear dependent upon each other when some vector can be found where each basis matrix can produce this vector through a linear combination of its own constituent basis vectors. For a very simple example, the mappings {{ket|{{map|5 8 12}} {{map|7 11 16}}}} and {{ket|{{map|7 11 16}} {{map|15 24 35}}}} are linearly dependent because both mappings contain the vector {{map|7 11 16}}. For a less obvious example, the mappings {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} and {{ket|{{map|1 2 3}} {{map|0 3 5}}}} are also linearly dependent, because the vector {{map|7 11 16}} can be found through linear combinations of each of their rows; in the first mapping's case, {{map|7 11 16}} = 7{{map|1 0 -4}} + 11{{map|0 1 4}}, and in the second mapping's case, {{map|7 11 16}} = 7{{map|1 2 3}} + -1{{map|0 3 5}}.
+A set of basis matrices are linear dependent upon each other when some vector can be found where each basis matrix can produce this vector through a linear combination of its own constituent basis vectors. For a very simple example, the mappings {{rket|{{map|5 8 12}} {{map|7 11 16}}}} and {{rket|{{map|7 11 16}} {{map|15 24 35}}}} are linearly dependent because both mappings contain the vector {{map|7 11 16}}. For a less obvious example, the mappings {{rket|{{map|1 0 -4}} {{map|0 1 4}}}} and {{rket|{{map|1 2 3}} {{map|0 3 5}}}} are also linearly dependent, because the vector {{map|7 11 16}} can be found through linear combinations of each of their rows; in the first mapping's case, {{map|7 11 16}} = 7{{map|1 0 -4}} + 11{{map|0 1 4}}, and in the second mapping's case, {{map|7 11 16}} = 7{{map|1 2 3}} + -1{{map|0 3 5}}.
-Sometimes basis matrices can share not just one basis vector, but multiple basis vectors. For example, the comma basis {{bra|{{vector|-30 19 0 0}} {{vector|-26 15 1 0}} {{vector|-17 9 0 1}}}} and the comma basis {{bra|{{vector|-19 12 0 0}} {{vector|-15 8 1 0}} {{vector|-6 2 0 1}}}} share both the vector {{vector|4 -4 1 0}} as well as the vector {{vector|13 -10 0 1}}:
+Sometimes basis matrices can share not just one basis vector, but multiple basis vectors. For example, the comma basis [{{vector|-30 19 0 0}} {{vector|-26 15 1 0}} {{vector|-17 9 0 1}}] and the comma basis [{{vector|-19 12 0 0}} {{vector|-15 8 1 0}} {{vector|-6 2 0 1}}] share both the vector {{vector|4 -4 1 0}} as well as the vector {{vector|13 -10 0 1}}:
 * {{vector|4 -4 1 0}} = {{vector|-26 15 1 0}} - {{vector|-30 19 0 0}}