| Line 3: | Line 3: | ||
''Vector Transformations:'' | ''Vector Transformations:'' | ||
| − | 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) |
| − | + | f:(<math>\left(\begin{array}{c}a_1\\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> | <math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math> | ||
| − | Linear Transformations: | + | ''Linear Transformations:'' |
| − | A function L:V->W is a linear transformation of V to W if the following are true: | + | 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) | (1) L(u+v) = L(u) + L(v) | ||
(2) L(c*u) = c*L(u) | (2) L(c*u) = c*L(u) | ||
| − | In other words, a linear transformation is a vector transformation that also meets (1) and (2). | + | In other words, a ''linear transformation'' is a ''vector transformation'' that also meets (1) and (2). |
Revision as of 16:26, 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\\5&6&7&8\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).
