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A ''vector transformation'' is a function that is performed on a vector. (i.e. f:V->W) | A ''vector transformation'' is a function that is performed on a vector. (i.e. f:V->W) | ||
| − | + | <math>f:\left(\begin{array}{c}a_1\\a_2\\.\\a_n\end{array}\right)-> \left(\begin{array}{c}b_1\\b_2\\.\\b_n\end{array}\right)</math> | |
<math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math> | <math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math> | ||
Revision as of 16:29, 14 December 2011
Linear Transformations and Isomorphisms
Vector Transformations:
A vector transformation is a function that is performed on a vector. (i.e. f:V->W)
$ f:\left(\begin{array}{c}a_1\\a_2\\.\\a_n\end{array}\right)-> \left(\begin{array}{c}b_1\\b_2\\.\\b_n\end{array}\right) $
$ \left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right) $
Linear Transformations:
A function L:V->W is a linear transformation of V to W if the following are true:
(1) L(u+v) = L(u) + L(v) (2) L(c*u) = c*L(u)
In other words, a linear transformation is a vector transformation that also meets (1) and (2).
