Which of the following sets are closed under scalar multipli

Which of the following sets are closed under scalar multiplication?

(i) The set of all vectors in R2 of the form (a,b) where a + 3b = 0.

(ii) The set of all 2×2 matrices whose trace is equal to 1.

(iii) The set of all polynomials in P2 of the form a0 + a1x + a2x2 where the product a0a1a2 0.

(A) (ii) only (B) (i) and (iii) only (C) (ii) and (iii) only (D) (iii) only (E) all of them (F) (i) only (G) (i) and (ii) only (H) none of them

Solution

to fulfill the condition of lock under multiplication of a scalar must fulfill that:

the result of any scalar product between any element set should result also an element of the set.

the answer is that they all meet the condition.

the answer is (E)