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Linear Algebra

I would really appreciate some help on these problems. I really need to understand how to do these proofs. So, please be detailed.

1. Let V be a vector space and F: V R a linear map. Let W be the subset of V consisting of all elements v such that F(v)=0. Assume that W V, and let be an element of V which does not lie in W. Show that every element of V can be written as a sum w + c , with some w in W and some number c.

2. In exercise (1), show that W is a subspace of V. Let { } be a basis of W. Show that { } is a basis of V.

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This solution is comprised of a detailed explanation to show that every element of V can be written as a sum w + c , with some w in W and some number c.

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