Question

Show that the transformation T defined by
T(x1, x2) = (2x1 - 3x2, X1+4, 5x2) is not linear.
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Answer 1

Let u = (1, 0) and v = (0, 1). Then

T (u) = (2*1 - 3*0, 1 + 4, 5*0) = (2, 5, 0)

T (v) = (2*0 - 3*1, 0 + 4, 5*1) = (-3, 4, 5)

=>   T (u) + T (v) = (-1, 9, 5)

but

T (u + v) = T (1, 1) = (2*1 - 3*1, 1 + 4, 5*1) = (-1, 5, 5)

=>   T (u + v) ≠ T (u) + T (v)

which means T does not preserve addition, so it is not linear.