Q:

Find the formula of linear transformation

Here I have such an exercise:
Let the linear transformation $T: R^3 \to R^3$ be defined this way:
$T(e_1)=2e_1+3e_2+5e_3$
$T(e_2+e_3)=e_1$
$T(e_1+e_2+e_3)=e_2-e_3$
, where $(e_1,e_2,e_3)$ is the canonical base of the following linear space: $(R^3,+,\cdot,R)$. Calculate $T(x)$ for any $x \in R$(find the formula for this linear transformation). Can you please help me solve this. It's the first time I encounter such an exercise. Thank you very much.

A:

A linear transformation $\;T\;$ as defined by you doesn't exist, since it must be that $$e_2-e_3=T(e_1+e_2+e_3)=Te_1+T(e_2+e_3)=3e_1+3e_2+5e_3\iff$$
$$\;\;\iff\;\;3e_1+2e_2+6e_3=0$$
and the last equality is impossible as $\;\{e_1,e_2,e_3\}\;$ is a basis and thus linearly independent

A:

Let $x = (a,b,c) = ae_1+be_2+ce_3 = me_1+n(e_2+e_3)+p(e_1+e_2+e_3)$. You then solve $m,n,p$ interms of $a,b,c$ and you are done.

