Question 417736

T(m*(a, b) + n*(c, d)) = T((m*a, m*b)+(n*c, n*d))
= T(m*a + n*c, m*b + n*d)
= (m*a + n*c,  m*b + n*d, m*a + n*c + m*b + n*d)
= (m*a, m*b, m*a + m*b) + (n*c, n*d, n*c + n*d)
= m*(a,b,a+b) + n*(c,d, c+d)
= m*T(a,b) + n*T(c,d).

Therefore T is a linear transformation.