Hello! I Would like a proof detailed of the following question. 

Let V=F[x] to be the set of all polynomials over the field F and W to be the set of these polynomials that have degree less than or equal to n. 
Find a basis of V/W and dimV/W. 

  • V would be an infinite dimensional space. x^n+1, x^n+2, x^n+3 and so on would be elements there.

  • I said:" I Would like a proof detailed of the following question"

Answer

Solution attached

  • How can you deduced that x^n+1, x^n+2, x^n+3,... is a basis for V/W?

  • Well, we have to think of what V/W is. After quotioning by W, all that is left are polynomials of degree larger than n. So we need every possible power of x with exponent larger than n.

  • Great! I realized that the representatives of each basis vector of V/W, that is, x^n+1, x^n+2,... are just the vectors that complete a basis of V. For finite-dimensional vector spaces, I know that the basis completion theorem can be applied; but I'm not sure if that theorem can be applied to a finite dimensional vector space.

The answer is accepted.