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My teacher has defined [itex]U_1 = \langle1, 0, 0\rangle[/itex], [itex]U_2 = \langle0, 1, 0\rangle[/itex], and [itex]U_3 = \langle0, 0, 1\rangle[/itex].

So it seems like the function maps [itex]L(\langle1, 0, 0\rangle, \langle0, 1, 0\rangle) = a, L(\langle1, 0, 0\rangle, \langle0, 0, 1\rangle) = b,[/itex], and [itex]L(\langle0, 1, 0\rangle, \langle0, 0, 1\rangle) = c[/itex]

I'm not sure how that helps me determine what [itex]L(\langle v_1, v_2, v_3\rangle, \langle w_1, w_2, w_3\rangle)[/itex] is.

Could the function perhaps be [itex]L(u, v) = L(\langle v_1, v_2, v_3\rangle, \langle w_1, w_2, w_3\rangle) = a\cdot{}v_1 + b\cdot{}w_3 + c\cdot{}v_2[/itex]

That seems to satisfy our initial conditions.

My textbook doesn't cover this and my teacher hasn't shown an example of how these functions work, so I'm unsure what to do.