# Linear transformation

#### Petrus

##### Well-known member
Hello MHB,
given a linear transformation F so that this is known
\displaystyle \left\{ \begin{aligned} \phantom{1}F(1,0,0)=(1,2,3) \\ F(1,1,0)=(0,0,1)\\ F(1,1,1)=(12,3,4)\\ \end{aligned} \right.
Decide F

progress:
$$\displaystyle F(e_1)=(1,2,3)$$
$$\displaystyle F(e_2)=F(e_1)+F(e_2)-F(e_1)=(0,0,1)-(1,2,3)=(-1,-2,-2)$$
$$\displaystyle F(e_3)=F(e_1)+F(e_2)+F(e_3)-(F(e_1)+F(e_2))= (12,3,4)-(-1,-2,-2)=(13,5,6)$$
My $$\displaystyle f(e_3)$$ is wrong acording to facit and I don't understand

Regards,
$$\displaystyle |\pi\rangle$$

#### Petrus

##### Well-known member
Hello MHB,
I have seen what my misstake was... I forgot that $$\displaystyle F(e_2)=(-1,-2,-2)$$ that means $$\displaystyle F(e_1)+F(e_2)=(0,0,1)$$ so $$\displaystyle F(e_3)=(12,3,3)$$ I was not thinking clear, I did confuse this in my brain but now I see!

Regards,
$$\displaystyle |\pi\rangle$$