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The composition of linear transformations is linear: if f : V -> W and g : W -> Z are linear, then so is g o f : V -> Z. In the finite dimensional case and if bases have been chosen, then the composition of linear maps corresponds to the multiplication of matrices. |
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The composition of linear transformations is linear: if f : V -> W and g : W -> Z are linear, then so is g o f : V -> Z. In the finite dimensional case and if bases have been chosen, then the composition of linear maps corresponds to the multiplication of matrices. |
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Occasionly, V and W can be considered as vector spaces over different ground fields, and it is then important to specify which field was used for the definition of "linear". If V and W are considered as spaces over the field K as above, we talk about K-linear maps. For example, the conjugation of complex numbers is an R-linear map C -> C, but it is not C-linear. |
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