Consider the set U = {f € Co (R)| limx→∞ f(x) = 0}. That is, U is the set of all continuous functions that go to zero as x goes to infinity. Is U a vector space? Check ALL 10 AXIOMS and clearly show which axioms U satisfies and which it does not. Then conclude whether or not U is a vector space. Include every step, if you ever ask yourself "should I also include this??" the answer is YES. Your solution must be clear, complete, and detailed. Hint: U is a set of functions. Don't confuse those functions with their limits.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.