Which of these transformations is not linear? The input is v = (v1, v2). (i) T(v) = (v1, v2). (ii) T(v) = (v1, v1). (iii) T(v) = (0, v1). (iv) T(v) = (0, 1).

Answer 1

A linear transformation should preserve addition and scalar multiplication. In addition to v = (v1, v2), let u = (u1, u2), and let c be a scalar. We should see:

1. T(u + v) = T(u) + T(v)
2. T(c*v) = c*T(v)

The fourth transformation fails to preserve either:

1. T(u + v) = (0,1) since everything is taken to (0,1) by this transformation. But T(u) + T(v) = (0,1) + (0,1) = (0,2). So, T(u + v) does not equal T(u) + T(v).
2. T(c*v) = (0,1) since the transformation takes everything to (0,1), but c*T(v) = c*(0,1) = (0,c). So T(c*v) does not equal c*T(v).