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Let P[x] denote the vector space of all polynomials. Let E be the set of polynomials which only
contain terms of even degree of even degree. Is E a subspace of P[x]? Explain.
Solution
Let E be the set as given in the problem
E = { f(x)| f(x) contains only terms of even degree}
= { g(x2)| g any polynomial in F[x]}
If g and h are any polynomials in F[x], clearly g+h and cg are polynomials in F[x] and
(g+h)(x2) and cg(x2) belong to E (for any scalar c in F).
So E is closed under vector addition and scalar multiplication. So E is a subspace of F

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Let P[x] denote the vector space of all polynomials. Let E be the set.pdf

  • 1. Let P[x] denote the vector space of all polynomials. Let E be the set of polynomials which only contain terms of even degree of even degree. Is E a subspace of P[x]? Explain. Solution Let E be the set as given in the problem E = { f(x)| f(x) contains only terms of even degree} = { g(x2)| g any polynomial in F[x]} If g and h are any polynomials in F[x], clearly g+h and cg are polynomials in F[x] and (g+h)(x2) and cg(x2) belong to E (for any scalar c in F). So E is closed under vector addition and scalar multiplication. So E is a subspace of F