Linear mapping/Introduction/Section
Let be a field, and let and be -vector spaces. A mapping
is called a linear mapping if the following two properties are fulfilled.
- for all .
- 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 -linear. The identity , the null mapping , and the inclusion of a linear subspace are the simplest examples of a linear mapping. For a linear mapping, the compatibility with arbitrary linear combination holds, that is,
see exercise.
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
Proof
Let be a field, and let and be -vector spaces. Let
be a bijective linear map. Then also the inverse mapping