Question

Let H be the set of third degree polynomials H = {ax + ax2 + ax aEC} Is H a subspace of P3? Why or why not? Select all correct answer choices (there may be more than one). 0 a. H is a subspace of P3 because it contains only second degree polynomials 1b. H is a subspace of P3 because it can be written as the span of a subset of P3 OcH is not a subspace of P3 because it is not closed under vector addition Od. H is not a subspace of P3 because it does not contain the zero vector of P3 e. H is a subspace of P3 because it is closed under vector addition and scalar multiplication Of. H is not a subspace of P3 because it is not closed under scalar multiplication 3. H is a subspace of P3 because it contains the zero vector of P3 0 n

Answer 1

a H=faz+ax +ax8! a ec} = {a(2+2422): aec} = (x+2°+2) Thus His written as span of {x+22723} Hence Hisa Subspace. Thus (b) is correct option, and (@) is cosong, C, d) (f) are also terong. Now for axtatadi?, but bx?, bx? Thus than som ga+buy+ ax2)+( baltbart bB) = (a+b)x+(a+b)a?+ 6+ b) az EH and for a EC d. lantaa2 + ax) = da ataan?zlan? EH Hence H is closed under vector addition and scalar multiplication thue Hirea subepace. Hence connect optionse (el and (9) le wrong since only containment of zero veeder may not imply that Hirea subepace Hence corted option ie (6) )