Determine if a set is a subspace.

Question

Determine whether the given set is a subspace of P3. Explain why. 

A = {p(t) = at^3+bt} Where a and b are both elements of the Reals. 

Answer 1

Look up in you linear algebra textbook the properties of a vector space and test to see if vectors of the form p(t) satisfy ALL of them under the operations of vector addition and multiplication by scalars. Under the operation of vector addition, you will have to satisfy closure, the commutative property, the associative property, the existence of a single additive identity for the entire set, and the existence of an additive inverse for each vector in the set. You will also need to demonstrate compatibility of scalar multiplication with field multiplication, the identity element of scalar multiplication, the distributive property of scalar multiplication with respect to vector addition, and the distributive property of scalar multiplication with respect to field addition.