(By the way, in Math is the standard denoting of the 3-dimensional projective space - not
the standard denoting for the space of polynomials of degree 3.
It is why I said that the post is nonsense.
Indeed, without explicit specification/specifications on what is , it is so - I firmly believe in it . . . )
I am only vaguely familiar with this subject; so my response might not be mathematically rigorous.
We are to show that every polynomial of degree 3 (element of P³) can be formed by a linear combination of , , , and .
Let an arbitrary third degree polynomial be . Show that the polynomial can be expressed as .
We need to show that we can express the arbitrary coefficients a, b, c, and d in terms of p, q, r, and s. It is easy to see that this is possible by equating the coefficients of each term in the two polynomials:
a = p; b = p+2q; c = r; d = q+3s
Example: (a = 7; b = -5; c = 4; d = -3)
a = p = 7;
b = p+2q = 7+2q = -5 --> q = -6;
c = r = 4;
d = q+3s = -6+3s = -3 --> s = 1.
