The set R[x]n = { f (x) ∈ R[x] | the degree of f (x) ≤ n} is a vector space over R under the usual addition and scalar multiplication operations. However, the set consisting of polynomials of degree n is not a vector space over R. Why ?
2026-03-26 01:11:35.1774487495
the set consisting of polynomials of degree n is not a vector space over R. Why
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Well, its not a vector space since your space is not closed under addition:
$(x^2+x+1) + (-x^2+2x+2) = 3x+3.$
You start from polynomials of degree 2. But the sum is a polynomial of degree 1.