Prove Independence

I'm pretty sure you mean, S = {1 + x², x + x³}

It would be helpful for you to state the complete problem. My guess is that it is two parts:
a) Prove that the functions in S = {1 + x², x + x³} are linearly independent.
b) Augment S to a set S' that is a basis for P₃.

For a, how is linear independence defined? From your work above, I'm not sure that you know. The definitions for linear independence and linear independence are similar, and there is a subtlety that students often don't grasp.
For b, have you learned about the Gram-Schmidt process?

 

Prob. 15 is almost identical to prob. 13. The polynomials in P₃ are essentially the same as vectors in R⁴. For example, 1 + 2x² <---> <1, 0, 2, 0>.

 

This isn't the definition, and besides, a definition of a term ought not use the same term in the definition. Look in your book and see how it defines linear independence.

This is a necessary condition for linear independence, but it is not sufficient. For example, consider the set {<1, 0, 0>, <0, 1, 0>, <1, 1, 0>}. None of these vectors is a multiple of any other vector in the set, yet these vectors are not linearly independent.

 

Yes.

 

You can use the augmented basis you found in #13, and "untransform" the vectors to get the other two polynomials you need for a basis for P₃.

 

Sure, those vectors are linearly independent, one of many possible sets of four vectors that span R^4.

 

Yes, but with just two vectors, that's overkill. Two vectors are linearly independent as long as neither one is a multiple of the other. If you have three vectors, though, it's not as obvious. I gave you an example of this in another thread.