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| − | '''Linear Transformations and Isomorphisms''' | + | <u>'''Linear Transformations and Isomorphisms'''</u> |
| − | + | <u>Vector Transformations:</u> | |
| − | A | + | A <u>vector transformation </u>is a function that is performed on a vector. (i.e. f:X->Y) |
| − | <math>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)</math> | + | <math>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)</math> |
| − | Where | + | Where <math>X = \left(\begin{array}{c}x_1\\x_2\\.\\.\\x_n\end{array}\right)</math> and <math>Y = \left(\begin{array}{c}y_1\\y_2\\.\\.\\y_n\end{array}\right)</math> |
| − | <math>X = \left(\begin{array}{c}x_1\\x_2\\.\\.\\x_n\end{array}\right)</math> | + | |
| − | and | + | |
| − | <math>Y = \left(\begin{array}{c}y_1\\y_2\\.\\.\\y_n\end{array}\right)</math> | + | |
| + | <br> | ||
| + | <u>Linear Transformations</u>'':'' | ||
| − | + | A function L:V->W is a <u>linear transformation </u>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 <u>linear transformation </u>is a <u>vector transformation </u>that also meets (1) and (2). | |
| − | (2) | + | |
| − | + | <br> | |
| + | <br> | ||
| + | <br> | ||
| − | + | <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> | + | |
[[Category:MA265Fall2011Walther]] | [[Category:MA265Fall2011Walther]] | ||
Revision as of 16:41, 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\\y_2\\.\\.\\y_n\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) $
