SOLUTION: Show that P³= span {x²+x³, x, 2x²+1,3}

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Question 1139652: Show that P³= span {x²+x³, x, 2x²+1,3}
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52775)   (Show Source): You can put this solution on YOUR website!
.

Makes no sense. 

    (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 . . . )


Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


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.
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