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    Computing the pullback $T^t f$ for three operators on $F^2$

    E-HK-37-1

    Area
    Algebra
    Topics
    Linear Algebra, Linear Functionals, Transpose Transformation
    Status
    draft

    Let \(F\) be a field and let \(f\) be the linear functional on \(F^2\) defined by \(f(x_1, x_2) = ax_1 + bx_2\). For each of the following linear operators \(T\), let \(g = T^t f\), and find \(g(x_1, x_2)\).

    1. \(T(x_1, x_2) = (x_1, 0)\);

    2. \(T(x_1, x_2) = (-x_2, x_1)\);

    3. \(T(x_1, x_2) = (x_1 - x_2,\; x_1 + x_2)\).

    Solution

    By definition, \[ (T^tf)(x)=f(Tx). \] Therefore:

    1. If \(T(x_1,x_2)=(x_1,0)\), then \[ g(x_1,x_2)=a x_1. \]

    2. If \(T(x_1,x_2)=(-x_2,x_1)\), then \[ g(x_1,x_2)=a(-x_2)+b x_1=b x_1-a x_2. \]

    3. If \(T(x_1,x_2)=(x_1-x_2,x_1+x_2)\), then \[ \begin{aligned} g(x_1,x_2) &=a(x_1-x_2)+b(x_1+x_2)\\ &=(a+b)x_1+(b-a)x_2. \end{aligned} \]

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