Kunze Linear Algebra · Problem 3.7.1

Linear Transformations

Mathematics — Methods · transpose-linear-functional

Problem

Let FF be a field and let ff be the linear functional on F2F^2 defined by f(x1,x2)=ax1+bx2f(x_1, x_2) = ax_1 + bx_2. For each of the following linear operators TT, let g=Ttfg = T^t f, and find g(x1,x2)g(x_1, x_2). (a) T(x1,x2)=(x1,0)T(x_1, x_2) = (x_1, 0); (b) T(x1,x2)=(−x2,x1)T(x_1, x_2) = (-x_2, x_1); (c) T(x1,x2)=(x1−x2,x1+x2)T(x_1, x_2) = (x_1 - x_2, x_1 + x_2).

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