Question

Suppose that T:P2->P3 is a linear transformation where T(1)=1+x^2
T(x)=x^2 -x^3
T(x^2)=2+x^3
obtain the bases for Im and Ker(T)


2 give me a minute to type

Answer 1

is my solution for number 1

Answer 2

Find the polynomial in P2 whose coordinate matrix with respect to the basis
\[B=\left\{ x+x ^{2},1+x \right\}\]
is


the last one
Let B be the natural basis for R^3 and B' be a second basis ,such that

is the matrix representing the vector (2,5,-1) relative to the basis B' without calculating B'