Polynomial Functions of a Linear Transformation
\[f(x)=2+3x\]
and \[T\]
is a linear transformation such that \[T(x,y)=(2x-y,x+y)\]
.If
\[h(x)=a_0 + a_1 x +a_2 x^2 + ...+a_n x^n\]
we can define \[h(T)=a_0 I + a_1 T +a_2 T^2 +...+a_n T^n\]
Hence
\[f(T)=(2I+3T)(x,y)=2(x,y)+3(2x-y,x+y)=(8x-3y,3x+y)\]
.If
\[g(x)=x+x^2\]
then\[\begin{equation} \begin{aligned} g(T)(x,y) &= (T+T^2)(x,y) \\ &= T(x,y)+T(T(x,y)) \\ &= (2x-y,x+y)+T(2x-y,x+y) \\ &= (2x-y,x+y) +(2(2x-y)-(x+y), 2x-y+x+y) \\ &= (5x-4y,4x+y \end{aligned} \end{equation}\]
