# How did they get this linear system?

While I was doing some bases excercises, I checked the solution and found that they got that linear system above from $$c_1 \cdot (x^2 -x+1) + c_2\cdot(2x+1)+c_3\cdot (2x-1)$$

However, when I tried it myself, I got this as a linear system $$\begin{cases} c_1= a_2\\ -c_1+2c_2+2c_3=a_1\\ c_1+c_2-c_3=a_0 \end{cases}$$

What am I doing wrong here?

Edit : Forgot to add $$a_0$$ in the third equation. Fixed.

• Your third line is not an equation. This is probably not right... – Yves Daoust Sep 25 '19 at 14:53
• Maybe you forgot "$=a_0$" in the third line? – user Sep 25 '19 at 14:54

Your are doing nothing wrong... The system you obtain is equivalent to the first one you mention, if you change the rhs, namely substituting $$a_1$$ by $$a_1+a_2$$ and $$a_0$$ by $$a_0-a_2$$. They probably just forgot to move $$c_1$$ to the rhs.
$$c_1\cdot x^2+(-c_1+2c_2+2c_3)x+(c_1+c_2-c_3)$$
$$\begin{cases} c_1= a_2\\ -c_1+2c_2+2c_3=a_1\\ c_1+c_2-c_3=a_0 \end{cases}$$