Question

Name: Math 23 6. (14 points) Determine whether the following subsets are subspaces of the given veeto r space. Either prove that the set is a subspace or prove that it is not (a) The subset T C Ps of polynomials of degree less than or equal to 3 that are of the form p(x)-1+iz+o2+caz3, where c,02, c3 are scalars in R. (b) The set s-a a,bERM22, that is, the subset of all 2 x 2 matrices A where a11-a22 and a21-a12

Answer 1

Subspace: W, is a subset of a vector space V (F) is a subspace of v iff W is itself a vector space over the same scalar field F. Also, W is a vector subspace of v iff a alpha + beta elementof W: forall alpha, beta elementof W < & a elementof f (a) The subset T lessthanorequalto p_3 (R), polynomials of degree less than or equal to 3 T = {p(x) elementof 3/p(x) = perp + c_1 x + c_2 x^2 + c_3 x^3: c_1, c_2, c_3 elementof R} forall a elementof R, and p, g elementof T, we have (a p + g) (x) a p(x) + g(x) = a(1 + c_1 x + c_2 x^2 + c_3 x^3) + (1 + b_1 x + b_2 x^2 + b_3 x^3) where p(x) = 1 + c_1 x + c_2 x^2 + c_3 x^3 elementof T & g(x) = 1 + b_1 x + b_2 x^2 + b_3 x^3 elementof T = (a + perp) + (a c_1 + b_1)x + (ac_2 + b_2)x^2 + (ac_3 + b_3)x^3 which need not belong to T as a elementof R implies a + 1