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The following linear transformation takes any vector in '''R'''<sup>''2''</sup> and maps it to another vector in '''R'''<sup>''2''</sup> of same length rotated 45 degrees counter clockwise. | The following linear transformation takes any vector in '''R'''<sup>''2''</sup> and maps it to another vector in '''R'''<sup>''2''</sup> of same length rotated 45 degrees counter clockwise. | ||
| − | <math> | + | <math>\ T(X)= \mathbf{A}X </math> |
| + | |||
| + | where <math> \mathbf{A} = \begin{bmatrix}cos(\pi/2) & sin(\pi/2) \end{bmatrix} </math> | ||
Revision as of 11:05, 11 September 2008
What Does Linearity Mean?
Linearity describes a special property of a transformation T from Rn to Rm such that any linear combination of inputs yields the respective linear combination of their outputs. A transformation such as this remains closed under the operations of addition and scalar multiplication.
Example of a Linear Transformation (system)
The following linear transformation takes any vector in R2 and maps it to another vector in R2 of same length rotated 45 degrees counter clockwise.
$ \ T(X)= \mathbf{A}X $
where $ \mathbf{A} = \begin{bmatrix}cos(\pi/2) & sin(\pi/2) \end{bmatrix} $
