Base change/Transformation matrix/Polynomial ring/Polynomials of degree 3/1/Exercise

Let ${}V$ be the vector space of all polynomials of degree ${}\leq 3$, with the basis ${}{\mathfrak {u}}=X^{0},X^{1},X^{2},X^{3}$. Show that the polynomials

$X^{3}+3X^{2}-X+4,\,-X^{3}+4X^{2}+5X+3,\,2X^{2}-X+1,\,3X^{3}+5X^{2}+7X-2$

form a basis of ${}V$, and determine the transformation matrices.

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