Let $T:\mathbb{P_2}\rightarrow\mathbb{P_2}$ be a linear transformation defined by $T(a+bt+ct^2)=3a+(5a-2b)t+(4b+c)t^2$ Let $T:\mathbb{P_2}\rightarrow\mathbb{P_2}$ be a linear transformation defined by $T(a+bt+ct^2)=3a+(5a-2b)t+(4b+c)t^2$

Find the basis C for $\mathbb{P_2}$ so that the matrix $[T]_c$ is a digonal matrix 

My attmept: Suppose we take basis of C is ${1, t, t^2}$ i am little confusing can any   help me to solve
You have to find a basis $C=\{p_1,p_2, p_3\}$ of $\mathbb{P_2}$ and real numbers $t_1,t_2,t_3$ such that

$T(p_j)=t_jp_j$ for  $j=1,2,3$.

Then: $[T]_c=diag(t_1,t_2,t_3)$ 

FRED
Do The following operations on the T matrix w.r.t standard basis. $ C'_2\,=  C_2 \, - \, 4. C_3\,\, \\\\\, C'_1 = \, 2. C_1-\,\,5.C'_2 $. to get a diagonal matrix. Do the same operation on standard basis matrix to obtain $ [2,\, 5+t,\,-20-4t+ t^2] $ as basis. This basis will give a diagonal matrix.