Linear Algebra, 2020a

Section IV. Jordan Form 463
2.38 In P n (R), the vector space (over the reals) of degree n polynomials,
E = {p(x) ∈ P n (R) | p(−x) =p(x) for all x}
and
O = {p(x) ∈ P n (R) | p(−x) =−p(x) for all x}
are the even and the odd polynomials; p(x) =x 2 is even while p(x) =x 3 is odd.
Show that they are subspaces. Are they complementary? Are they invariant under
the differentiation transformation?
2.39 A matrix S is the square root of another T if S 2 = T. Show that any nonsingular
matrix has a square root.