Linear mapping/Introduction/Section

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Let be a field, and let and be -vector spaces. A mapping

is called a linear mapping, if the following two properties are fulfilled.

  1. for all .
  2. for all and .

Here, the first property is called additivity and the second property is called compatibility with scaling. When we want to stress the base field, then we say -linearity. The identity , the null mapping and the inclusion of a linear subspace are the simplest examples for linear mappings.

Let denote a field, and let be the -dimensional standard space. Then the -th projection, this is the mapping

is a -linear mapping. This follows immediately from componentwise addition and scalar multiplication on the standard space. The -th projection is also called the -th coordinate function.

Let denote a field, and let denote vector spaces over . Suppose that

are linear mappings. Then also the composition

is a linear mapping.


Let be a field, and let and be -vector spaces. Let

be a bijective linear map. Then also the inverse mapping

is linear.