Linear Algebra Let V be the set of all real polynomials p wi

[Linear Algebra] Let V be the set of all real polynomials p with p(1) = 1. Is this a subspace of all the polynomials? Please explain!

Solution

Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if W satisfies the following three conditions:

No, if V be the set of all real polynomials p with p(1) = 1. this is not a subspace of all the polynomials,

consider a polynomial p = x ,now p(1) = 1 ,

now let us take c = 4 then c p(1) = c = 4    1 ,

hence cp is not element of V, (it does not satisfy property number 3 )

hence it is not subspace.