Ordered basis proof

Question

If e1,e2, e3 and f1 f2 f3 are ordered basis of R^3 I was able to prove that

T(x1e1 + x2e2 + x3e3) = x3f1 + x1f2 + x2f3,

Is a linear map. But what if I added x1 x2 x3

S(x1e1 + x2e2 + x3e3) = x2x3f1 + x1x3f2 + x1x2f3.

Would this also be a linear map, and if so how would I prove it

Answer 1

Extended hint from what I already gave in the comments:

A few things written in the format of $x_1\cdot e_1+ x_2\cdot e_2+x_3\cdot e_3$

$x_1\cdot e_1+ x_2\cdot e_2+x_3\cdot e_3$

$1~\cdot e_1~+0\cdot ~e_2~+0\cdot e_3=e_1$

$0~\cdot e_1~+1\cdot ~e_2~+0\cdot e_3=e_2$

$1~\cdot e_1~+1\cdot ~e_2~+0\cdot e_3=e_1+e_2$

So then, we ask the question, what is $S(e_1)$? What is $S(e_2)$? What is $S(e_1)+S(e_2)$? What is $S(e_1+e_2)$?

$S(e_1)=S(1\cdot e_1+0\cdot e_2+0\cdot e_3)=0\cdot 0\cdot f_1+1\cdot 0\cdot f_2+1\cdot 0\cdot f_3=0+0+0=0$