Finding parameters that give a linear transformation

Question

Let $L: \textbf{R}^3 \rightarrow \textbf{R}^3$ be defined by $L([x,y,z]) = [ax^2 + bx, cy + z, d]$.

Which of the following choices of the parameters $a, b, c, d$ gives a linear transformation?

A. $a = 1, b = 2, c = 3, d = 0$

B. $a = 0, b = 1, c = 0, d = 1$

C. $a = 0, b = 2, c = 4, d = 0$

D. $a = 1, b = 0, c = 0, d = 1$

E. $a = b = c = d = 1$

The solutions I have looked at online all say C is the answer because a and d need to be equal to zero, but none of them explain it. Why do $a$ and $d$ need to equal zero?

Answer 1

Note that

• $a$ is on a non linear term

• take $d\neq 0$ and check $L(u+v)=L(u)+L(v)$

Answer 2

Every component of $L(x,y,z)$ must be a linear combination of the variables $x$, $y$ and $z$. So you must have $a = d =0$, while $b$ and $c$ can be anything.