Difference between revisions of "HW2.C Adrian Delancy ECE301Fall2008mboutin" - Rhea

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What Does Linearity Mean?

Linearity describes a special property of a transformation T from Rⁿ to R^m such that any linear combination of inputs yields the linear combination of their respective 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. Using the standard basis vectors:

$ \ T(X)= \mathbf{A}X $

where $ \mathbf{A} = \begin{bmatrix}cos(\pi/2) & sin(\pi/2)\\ -sin(\pi/2) & cos(\pi/2) \end{bmatrix} $

and $ \ X $ is any vector in R²

Therefore $ \ T(C_1 e_1)= \begin{bmatrix}cos(\pi/2) & sin(\pi/2)\\ -sin(\pi/2) & cos(\pi/2)\end{bmatrix} \begin{bmatrix} C_1 \\ 0 \end{bmatrix} = \begin{bmatrix}C_1/ \sqrt(2) \\ -C_1/\sqrt(2) \end{bmatrix} $

and

$ \ T(C_2 e_2)= \begin{bmatrix}cos(\pi/2) & sin(\pi/2)\\ -sin(\pi/2) & cos(\pi/2)\end{bmatrix} \begin{bmatrix} 0 \\ C_2 \end{bmatrix} = \begin{bmatrix}C_2/ \sqrt(2) \\ C_2/\sqrt(2) \end{bmatrix} $

Now summing the two vectors:

$ \ Y = T(C_1 e_1) + T(C_2 e_2) = \begin{bmatrix}\frac{C_1} {\sqrt(2)} + \frac{C_2}{\sqrt(2)} \\ \frac{-C_1}{\sqrt(2)} + \frac{C_2}{\sqrt(2)} \end{bmatrix} $

Now summing the two vectors before putting them into the transformation:

$ \ X= C_1 e_1 + C_2 e_2 = \begin{bmatrix} C_1 \\ C_2 \end{bmatrix} $

$ \ T(X)= \begin{bmatrix}cos(\pi/2) & sin(\pi/2) \\ -sin(\pi/2) & cos(\pi/2)\end{bmatrix} \begin{bmatrix} C_1 \\ C_2 \end{bmatrix} $

$ \ T(X) = C_1 \begin{bmatrix} cos(\pi/2) \\ -sin(\pi/2) \end{bmatrix} + C_2 \begin{bmatrix}sin(\pi/2) \\ cos(\pi/2) \end{bmatrix} = \begin{bmatrix}\frac{C_1} {\sqrt(2)} + \frac{C_2}{\sqrt(2)} \\ \frac{-C_1}{\sqrt(2)} + \frac{C_2}{\sqrt(2)} \end{bmatrix}= T(C_1 e_1)+T(C_2 e_2) = Y $

Therefore the transformation is linear.

Example of a Non-Linear Transformation

Let a physical system be modeled by the following non-linear transformation:

$ \ T(x,y)= x\ln(y) $ Proof that it is not linear:

$ \ T(x_1, y_1) + T(x_2,y_2) = x_1 \ln(y_1)+ x_2 \ln(y_2) = U $

Now Summing the input signals and then putting it into the transformation yields:

$ \ T(x_1+x_2,y_1+y_2)= (x_1+x_2)\ln(y_1+y_2)=x_1 \ln(y_1+y_2) + x_2 \ln(y_1+y_2) = W $

$ \ U \ne W $ Therefore, transformation is not linear.