Answer to Question #87894 in Linear Algebra for muhasin

Question

check that p^(e)andp^(0) are subspace of R[x] if p^(e)={p(x)€R[x]|p(x)=p(-x) and p^(0)={p(x)€R[x]p(x)=-p(-x)

Answer 1

These sets p^(e), p^(0) are not empty.

Let f(x), g(x) € p^(e), a,b  € R, h(x) = af(x) + bg(x), then

h(x) = af(x) + bg(x) = af(-x) + bg(-x) = h(-x) therefore h(x) € p^(e) and p^(e) is subspace of R[x].

Let f(x), g(x) € p^(o), a,b  € R, h(x) = af(x) + bg(x), then

h(x) = af(x) + bg(x) = -af(-x) - bg(-x) = - ( af(-x) + bg(-x) ) = - h(-x) therefore h(x) € p^(o) and p^(o) is subspace of R[x].