b$)$ Use coordinate vectors $($the column$-$matrix representation of polynomials$)$ to verify that the set $=\{1-t^2,t-t^2,2-2t+t^2\}$ is a basis for $P_2($the set of all polynomials of degree at most two$).$ Also, find the coordinate vector of

$p(t)=6+3t-t^2$

relative to the basis $.$ What is $dim{P}_2=?.$ Justify your answer.