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A vector space is a set equipped with an operation called vector addition and with a given meaning for “multiplication by a scalar” (multiplication of an element by a number) subject to the following axioms:
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A linear subspace of a vector space is a subset of the vector space having the following properties:
The sum of any two members of the subset is in the subset.
The element of the vector space is in the subset.
The multiple of any element of the subset by any scalar is in the subset.
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A map from a vector space to a vector space is a linear map (or linear transformation) if it preserves linear combinations. This means that it satisfies the following axioms:
for all in
for all scalars and all in
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A map from a vector space to itself is called a translation if it has the form for some fixed in . The particular translation given by an element of is called translation by .
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