Let P 2 be the vector space of all polynomials of degree less than or equal to 2, under the usual vector addition and scalar multiplication. Let S = {f (x) ∈ P 2 | if deg(f ) = 2, then f has two complex roots}.  (a) Is "0 ∈ S. Why or why not? (b) Is S closed under addition? If S is closed under addition, prove it. If S is not closed under addition, provide a counterexample.   (c) Is S closed under scalar multiplication? If S is closed under scalar multiplication, prove it. If S is not closed under scalar multiplication, provide a counterexample.

(c) A line in R not through the origin ax t by =C #0 then ŏ ¢ H. :. tt s not asubspacd y = 9. Umecersung but ealightenng: X = x You could wnite this as a Sobset, H, of R? H» { ] kek Is H closed under ☺ ? Is H closed under ☺ ļ

3: Which of the following sets are closed under scalar multiplication? (1) The set of all vectors in R of the form (a, b, 9ab). (i1) The set of all polynomials a- bx+ cx in P, such that ac =D0. (iii) The set of all matrices in Mn such that A = A (A) (1) and (ii) only (B) (1) and (iii) only (C) (ii) and (iii) only (D) (1) only (E) (ii) only (F) none of them (G) (iii) only (H) all of them

Let A = Let a 7 -1 2152 -1 203 13350 where a is a scalar. Show that det(A) is independent of a.