Linear Transformations and Isomorphisms
Vector Transformations:
A vector transformation is a function that is performed on a vector. (i.e. f:X->Y)
A vector transformation can transform a vector from Rn to Rm
$ f:\left(\begin{array}{c}x_1\\x_2\\.\\.\\a_n\end{array}\right)-> \left(\begin{array}{c}y_1\\y_2\\.\\.\\y_m\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\\y_2\\.\\.\\y_m\end{array}\right) $
Examples:
1.
$ f(\left(\begin{array}{c}x_1\\x_2\end{array}\right))= \left(\begin{array}{c}x_1^2\\0\end{array}\right) $
$ X=\left(\begin{array}{c}1\\-2\end{array}\right) $
$ f(\left(\begin{array}{c}x_1\\x_2\end{array}\right))= \left(\begin{array}{c}-x_1\\x_1 - x_2\\x_1\end{array}\right) $
$ X=\left(\begin{array}{c}-1\\4\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) $
