Linear mapping/Determination theorem/Finite-dimensional/Section

From Wikiversity
Jump to navigation Jump to search

Behind the following statement (the determination theorem), there is the important principle that in linear algebra (of finite-dimensional vector spaces), the objects are determined by finitely many data.

Let be a field, and let and be -vector spaces. Let , , denote a basis of , and let , , denote elements in . Then there exists a unique linear mapping



This proof was not presented in the lecture.

The graph of a linear mapping from to , the mapping is determined by the proportionality factor alone.

The easiest linear mappings are (beside the null mapping) the linear maps from to . Such a linear mapping

is determined (by fact, but this is also directly clear) by , or by the value for an single element , . In particular, , with a uniquely determined . In the context of physics, for , and if there is a linear relation between two measurable quantities, we talk about proportionality, and is called the proportionality factor. In school, such a linear relation occurs as "rule of three“.