Linear Transformations and Isomorphisms<u</u>
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) $
Example 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}-1\\-2\end{array}\right)= \left(\begin{array}{c}-1^2\\0\end{array}\right)= \left(\begin{array}{c}1\\0\end{array}\right) $
Example 2:
$ 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) $
$ f(\left(\begin{array}{c}-1\\4\end{array}\right))= \left(\begin{array}{c}-(-1)\\-1 - 4\\-1\end{array}\right)= \left(\begin{array}{c}1\\- 5\\-1\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) denoted from now on as L:V ->W
Let's return to examples 1 and 2 to see if they are linear transformations.
Example 1:
We must check conditions (1) and (2)
(1):
$ L(\left(\begin{array}{c}u_1\\u_2\end{array}\right))= \left(\begin{array}{c}u_1^2\\0\end{array}\right) $
$ V=\left(\begin{array}{c}v_1\\v_2\end{array}\right)=\left(\begin{array}{c}2\\5\end{array}\right) $
$ L(\left(\begin{array}{c}u_1 + v_1\\u_2 + v_2\end{array}\right))= \left(\begin{array}{c}(u_1 + v_1)^2\\0\end{array}\right) $
$ L(\left(\begin{array}{c}-1 + 2\\-2 + 5\end{array}\right))= \left(\begin{array}{c}(-1 + 2)^2\\0\end{array}\right)= \left(\begin{array}{c}1\\0\end{array}\right) $
$ L(\left(\begin{array}{c}-1\\-2\end{array}\right))= \left(\begin{array}{c}-1^2\\0\end{array}\right)= \left(\begin{array}{c}1\\0\end{array}\right) $
$ L(\left(\begin{array}{c}-1\\-2\end{array}\right)= \left(\begin{array}{c}(-1)^2\\0\end{array}\right)+L(\left(\begin{array}{c}2\\5\end{array}\right))= \left(\begin{array}{c}(2)^2\\0\end{array}\right) $
$ \left(\begin{array}{c}1\\0\end{array}\right)+\left(\begin{array}{c}4\\0\end{array}\right)=\left(\begin{array}{c}5\\0\end{array}\right) $
and,
$ \left(\begin{array}{c}5\\0\end{array}\right)NOT=\left(\begin{array}{c}1\\0\end{array}\right) $
Therefore since,
$ L(\left(\begin{array}{c}u_1 + v_1\\u_2 + v_2\end{array}\right))NOT= L(\left(\begin{array}{c}u_1\\u_2\end{array}\right))+ L(\left(\begin{array}{c}v_1\\v_2\end{array}\right)) $
$ L(\left(\begin{array}{c}u_1\\u_2\end{array}\right))= \left(\begin{array}{c}u_1^2\\0\end{array}\right) $
is NOT a linear transform
