Show that the transformation T defined by T(x1, x2, x3) = {x1, x2, −2x3} is linear.

Question

Show that the transformation T defined by T(x₁, x₂, x₃) = {x₁, x₂, −2x₃} is linear.

Answer 1

Let v = <a, b, c> and w = <d, e, f>

T is linear if T(v + w) = T(v) + T(w) and T(kv) = kT(v), where k is any real number

T(v + w) = T(<a+d, b+e, c+f>) = < a+d, b+e, -2(c+f) > = <a, b, -2c> + <d, e, -2f> = T(v) + T(w)

T(kv) = T(<ka, kb, kc>) = <ka, kb, -2kc> = k<a, b, -2c> = kT(v)

T is a linear transformation since T(v + w) = T(v) + T(w) and T(kv) = kT(v)