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. 

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

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

Answer

Solution attached

Mathe Mathe
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Matchmaticians Hello! I Would like a proof detailed of the following question.  File #1 File #1 (pdf)
  • Homeomorfo Fle Homeomorfo Fle
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    How can you deduced that x^n+1, x^n+2, x^n+3,... is a basis for V/W?

  • Mathe Mathe
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    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.

  • Homeomorfo Fle Homeomorfo Fle
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    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.
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