Linear algebra formalism: Difference between revisions
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Note that this is the fundamental definition of what it means for something to be "a vector"; vectors are defined as objects in spaces where these axioms apply. | Note that this is the fundamental definition of what it means for something to be "a vector"; vectors are defined as objects in spaces where these axioms apply. | ||
Additionally, the axioms of linear algebra contain the axioms of group theory, so that the just intervals under stacking can be considered a group. | |||
Note that what we've described as multiplication is actually vector addition, and what we've described as exponentiation is actually multiplication of a vector (the interval) by a scalar (the exponent). Additionally, the unison is actually a zero vector. This makes sense if we think of intervals logarithmically, where multiplication of ratios becomes addition of [[cent]] values, the unison is 0 cents, and exponents become scale factors. | Note that what we've described as multiplication is actually vector addition, and what we've described as exponentiation is actually multiplication of a vector (the interval) by a scalar (the exponent). Additionally, the unison is actually a zero vector. This makes sense if we think of intervals logarithmically, where multiplication of ratios becomes addition of [[cent]] values, the unison is 0 cents, and exponents become scale factors. | ||