<s>[INST] Linear Transformations, Linear Algebra Let T:P3→P3 be the linear transformation such that $T(−2x^2)= −2x^2 − 2x$, $T(0.5x + 2)= 3x^2 + 4x−2$, and $T(2x^2 − 1)= 2x + 1$. 
Find $T(1), T(x), T(x^2)$, and $T(ax^2 + bx + c)$, where a, b, and c are arbitrary real numbers.
I understand how to find $T(x^2)$ where you just divide the given $T(-2x^2)$ by $-2$ to get $T(x^2) = x^2 + x$. 
I'm not sure sure how to proceed in calculating the other transformation functions. Please list as many of the steps as possible in solving for the three other functions.
Thank you!
 [/INST] We can write $1=-(-2x^2)-(2x^2-1), x=2(0.5x+2)-4$, so 
$$T(1)=T(-(-2x^2)-(2x^2-1))=-T(-2x^2)-T(2x^2-1)=2x^2-1$$
$$T(x)=T(2(0.5x+2)-4)=2T(0.5x+2)-4T(1)=-2x^2+8x$$
Then we get 
$$T(ax^2+bx+c)=a(x^2+x)+b(-2x^2+8x)+c(2x^2-1)$$
$$=(a-2b+2c)x^2+(a+8b)x-c$$
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