Val: Difference between revisions
→Vals vs. mappings: clarified a few things and also removed "homomorphic" as mapping matrices are also homomorphisms, just to Z^n rather than to Z |
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== Vals vs. mappings == | == Vals vs. mappings == | ||
A val is more specific than a [[mapping]] | A val is more specific than a [[mapping]]. To be precise: | ||
# A val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]], called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]]. This means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector). Or, if interpreted as a [[Wikipedia:Matrix_(mathematics)|matrix]], a val has only one row. | # A val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]], called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]]. This means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector). Or, if interpreted as a [[Wikipedia:Matrix_(mathematics)|matrix]], a val has only one row. | ||
# To be precise, the rows of integer mapping matrices are vals, so that mapping matrices can be thought of as being built up from vals. | |||
# It has only integer entries. | # It has only integer entries. | ||
# Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear sums of [[Wikipedia:P-adic_order|p-adic valuations]]. | # Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear sums of [[Wikipedia:P-adic_order|p-adic valuations]]. | ||