Linear transformations

Apprentice123 · #1 $Old$ November 14th, 2008, 05:24 PM

check that it is linear Transformation

$T:R^2 =>M(2,2)$

$T(x,y)=\begin{pmatrix} 2y & 3x \\ -y & x+2y\end{pmatrix}$

Scopur · #2 $Old$ November 15th, 2008, 02:03 PM

Quote:

Originally Posted by Apprentice123 $View Post$

check that it is linear Transformation

$T:R^2 =>M(2,2)$

$T(x,y)=\begin{pmatrix} 2y & 3x \\ -y & x+2y\end{pmatrix}$

if T is a linear transformation then it must be closed under addition (T1) and scalar multiplication (T2).

For T1. $T(x,y + x_1,y_1) = \begin{pmatrix} 4y & 6x \\ -2y & 2x+4y\end{pmatrix} =$

$\begin{pmatrix} 2y & 3x \\ -y & x+2y\end{pmatrix} + \begin{pmatrix} 2y & 3x \\ -y & x+2y\end{pmatrix} = T(x,y) + T(x_1, x_2)$

For T2. $T(kv) = \begin{pmatrix} 2yk & 3xk \\ -yk & k(x+2y)\end{pmatrix} = k\begin{pmatrix} 2y & 3x \\ -y & x+2y\end{pmatrix}= kT(v)$

Therefore the transformation T is a linear transformation.