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<math> \mathbf{y}[b] = \begin{bmatrix}7 & 12 \end{bmatrix} </math>, | <math> \mathbf{y}[b] = \begin{bmatrix}7 & 12 \end{bmatrix} </math>, | ||
<math> \mathbf{b} = \begin{bmatrix}4 & 1 \end{bmatrix} </math>, and | <math> \mathbf{b} = \begin{bmatrix}4 & 1 \end{bmatrix} </math>, and | ||
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<math> \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} </math> | <math> \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} </math> | ||
Revision as of 09:21, 10 September 2008
Linear System Definition
A system takes a given input and produces an output. For the system to be linear it must preserve addition and multiplication. In mathematical terms:
$ x(t+t0)=x(t) + x(t0) $
and
$ x(k*t)=k*x(t) $
Linear System Example
Consider the system $ \mathbf{y}[b]=\mathbf{b}\cdot\mathbf{M} $ where $ \mathbf{y}[b] = \begin{bmatrix}7 & 12 \end{bmatrix} $, $ \mathbf{b} = \begin{bmatrix}4 & 1 \end{bmatrix} $, and $ \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} $
let $ \mathbf{a} = \begin{bmatrix}2 & 2 \end{bmatrix} $
If the system is linear these properties hold:
$ y[a+b]=f[a]+f[b] \, $
$ y[kb]=ky[b] \, $
