Linear algebra

Question

Consider all polynomials that are zero at the real origin; that is, all polynomials such that p(0) = 0. Does this
create a subspace of set of all polynomials?

Answer 1

Yes it is a subspace.

 

To show it is a subspace, you must show 3 things.

a) Your subset is not the empty set

b)closure in addition

c) closure in scalar multiplication.

 

Let P(x) and Q(x) be polynomials that pass through (0,0)

 

a) this is certainly not empty since p(x) = x² is in the subset.

b) If P and Q are both in the set, then P+Q is also in the set since P(0) + Q(0) = 0+0 = 0 and so their addition still goes through (0,0).

c) if P is in the subset, the a*P(x) is in the subset since a*P(0) = a*0 = 0 so it still goes through (0,0).

 

So this is a subspace of the vector space of the set of all polynomials.

 

Hope this helped.