From Advanced Calculus , 5th Edition, by Wilfred Kaplan Section 1.18, problem #1

Question

From Advanced Calculus, 5th Edition, by Wilfred Kaplan Section 1.18, problem #1(a), pg 70...

1. Show that each of the following sets of objects, with the usual operations of addition and multiplication by scalars, forms a vector space. Give the dimension in each case and, if the dimension is finite, give a basis.

a) All polynomials of degree at most 2.

...I've read and re-read the chapter, along with viewing related Khan Academy videos...but I'm still not sure how to start on answering this question (and all sub-questions; (b) thru (l)). If I had an example, I think I could figure the rest out.

Explanation / Answer

Well start with the definition of a vector space, you have to have two binary operations, ie vector addition, and scalar multiplication. Don't worry too much about the exact meaning of them being binary operations if you're not in a mathematical algebra class, just worry about the 8 things you need to prove (copying list from wikipedia for brevity):' Associativity of addition v1 + (v2 + v3) = (v1 + v2) + v3. Commutativity of addition v1 + v2 = v2 + v1. Identity element of addition There exists an element 0 ? V, called the zero vector, such that v + 0 = v for all v ? V. Inverse elements of addition For all v ? V, there exists an element -v ? V, called the additive inverse of v, such that v + (-v) = 0. Distributivity of scalar multiplication with respect to vector addition s(v1 + v2) = sv1 + sv2. Distributivity of scalar multiplication with respect to field addition (n1 + n2)v = n1v + n2v. Respect of scalar multiplication over field's multiplication n1 (n2 s) = (n1 n2)s Identity element of scalar multiplication 1s = s, where 1 denotes the multiplicative identity in F. So we basically need to show all 8 are true for some identity element, and some set of scalars/vectors, that we want to use. There are many ways to start doing this. The best IMO is to realize the following: the set of all polynomials of degree

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