Linear transformations
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Linear Transformations
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[Equation goes here - download the original to see it.] We can extend this concept of a structure preserving mapping between vector spaces to encompass the case where the mapping is not necessarily one - one, and also where the vector space is not necessarily finite. We call this a linear transformation. If V and W are vector spaces (either finite or infinite dimensional) then a function [Equation goes here - download the original to see it.] is a linear transformation if for all vectors [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] The two statements can be combined into a single expression [Equation goes here - download the original to see it.] Example: Show that the process of differentiation defines a linear transformation from the vector space of all polynomials onto itself. Solution: Let V be the set of all polynomials. Then any element of [Equation goes here - download the original to see it.] has the form [Equation goes here - download the original to see it.] for some integration. (Since n can be as large a number as one pleases this vector space is actually infinite dimensional) Differentiation of this element gives [Equation goes here - download the original to see it.] This is also an element of V, so differentiation can be regarded as an operator mapping V to V. We can symbolise it as a many - valued function D: [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.]
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Contents of
Linear transformations
1 Linear Transformations
2 Finite Linear Transformations
3 Composition of Linear Transformations
4 Kernels and Images
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