In K4[x], the linear space of the polynomials of degree 4 or less with coefficients in in K, determine:

(a) A basis that doesn't contain 2nd or 3rd degree polynomials

(b) If there exists a basis that doesn't contain 4th degree polynomials

(c) A supplementary subspace of the one generated by the polynomials p(x)=−1+2x+x^2 y q(x)=1+x+x^2

Suggestion: use coordinate row (or column) matrices

  • $\begingroup$ Hi Chachi Kent! Welcome to MSE. It is very helpful to potential answerers if you format your post using MathJax. That way it's easy to read and thus easier to answer $\endgroup$ – eepperly16 Dec 12 '17 at 20:00
  • $\begingroup$ What are your thoughts? Do you understand what the "suggestion" is saying? $\endgroup$ – angryavian Dec 12 '17 at 20:23
  • $\begingroup$ That the coefficients of the polynomial(s) would form the rows/columns of a matrix, right? But if there's no 2nd or 3rd degree, two whole rows of the martix would be null or would just two elements of it be null? $\endgroup$ – Chachi Kent Dec 13 '17 at 9:45

The basis for a 4th-degree polynomial is {1,x,x^2,x^3,x^4}. So I think the answer to part (a) is {1,x,x^4}. For part (b) there is not.


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