Can you please help with this linear algebra problem? Please do not use the answer on other websites since I believe it is incorrect. Thank you!

Let V be the set of realvalued functions that are dened at each 1: in the interval ( oo, 00). If f = f (I) and g = g(;r) are two functions in V and k is any scalar, we dene the operations of addition and scalar multiplication by (f + am : rm + gm, (the) = we). Verify the Vector Space Axioms for the given set of vectors. Vector Space Axioms The following denition consists MW eight of which are properties of vectors in R\" that were stated in Theorem 3.1.1. It is important to keep in mind that one does not prove axioms; rather, they are assumptions that serve as the starting point for proving theorems. DEFINITION 1 Let V be an arbitrary nonempty set of objects on which two operations are dened: addition, and multiplication by numbers called scalars. By addition we mean a rule for associating with each pair of objects 11 and v in V an object u + v, called the sum of u and v; by scalar multiplication we mean a rule for associating with each scalar k and each object u in V an object ku, called the scalar multiple of u by k. If the following axioms are satised by all objects u, v, w in V and all scalars k and m, then we call V a vector space and we call the objects in V vectors. 1. 2. 3. 'J I 10. Ifu and vare objects in V, thenu+vis in V. u+v=v+u u+(v+w):(u+v)+w There is an object 0 in V, called a zero vector for V, such that 0 I u = u + 0 = u for all u in V. For each u in V, there is an object u in V, called a negative of u, such that u+(u) : (u)+u:0. Ifk is any scalar and u is any object in V, then ku is in V. k(u+v) : ku+kv (k + m)u : ku -I mu k(mu) : (km)(u) luzu