We are looking for
(Be careful! The operations used here are tricky.1)
Key idea
The existence of such a non-zero
is equivalent to the following: the transformation has a non-zero kernel, and, therefore, is not invertible. This is in turn equivalent to
for a basis in . (We use here the observation that .)
(While
Footnotes
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is a multiplication result of a scalar by a transformation (which multiplies all values of the transformation by the scalar), whereas is a subtraction of transformations (once more, for each vector the result of applying is ) ↩