Answer to Question #143174 in Linear Algebra for Sourav Mondal

Question #143174

Let T : R3 -> R3 be the linear operator defined by T(x1, x2, x3) = (x1, x3, -2x2- x3).

Let f(x) = - x³+ 2.

Find the operator f(T)


1
Expert's answer
2020-11-12T18:44:22-0500

Given T[x1x2x3]=[x1x32x2x3]T\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} x_1 \\ x_3 \\ -2x_2-x_3 \end{bmatrix}


In matirx form we have T[x1x2x3]=[100001021][x1x2x3]=AXT\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} =\begin{bmatrix} 1 &0 &0 \\ 0 & 0& 1 \\ 0 & -2 & -1 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = AX


Where A=[100001021]A=\begin{bmatrix} 1 &0 &0 \\ 0 & 0& 1 \\ 0 & -2 & -1 \end{bmatrix} ; X=[x1x2x3]X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}


and f(x)=x3+2f(x) = -x^3+2


then f(T)=f(A)=A3+2If(T)= f(A)=-A^3+2I


A3=[100021023]A^3=\begin{bmatrix} 1 &0 &0 \\ 0 & 2& -1 \\ 0 & 2 & 3 \end{bmatrix} , 2I=2[100010001]=[200020002]2I =2\begin{bmatrix} 1 &0 &0 \\ 0 & 1& 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 &0 &0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}


f(A)=[100021023]+[200020002]=[100001021]=Af(A)=\begin{bmatrix} -1 &0 &0 \\ 0 & -2 & 1 \\ 0 & -2 & -3 \end{bmatrix} + \begin{bmatrix} 2 &0 &0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} = \begin{bmatrix} 1 &0 &0 \\ 0 & 0& 1 \\ 0 & -2 & -1 \end{bmatrix} =A


Therefore f(T)=Tf(T)=T


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