| Line 10: | Line 10: | ||
<math>X = \left(\begin{array}{c}x_1\\x_2\\.\\.\\x_n\end{array}\right)</math> | <math>X = \left(\begin{array}{c}x_1\\x_2\\.\\.\\x_n\end{array}\right)</math> | ||
and | and | ||
| − | <math> Y = \left(\begin{array}{c}y_1\\x_y\\.\\.\\x_y\end{array}\right)</math> | + | <math>Y = \left(\begin{array}{c}y_1\\x_y\\.\\.\\x_y\end{array}\right)</math> |
Revision as of 16:39, 14 December 2011
Linear Transformations and Isomorphisms
Vector Transformations:
A vector transformation is a function that is performed on a vector. (i.e. f:X->Y)
$ f:\left(\begin{array}{c}x_1\\x_2\\.\\.\\a_n\end{array}\right)-> \left(\begin{array}{c}y_1\\y_2\\.\\.\\y_n\end{array}\right) $
Where $ X = \left(\begin{array}{c}x_1\\x_2\\.\\.\\x_n\end{array}\right) $ and $ Y = \left(\begin{array}{c}y_1\\x_y\\.\\.\\x_y\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).
$ \left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right) $
