1. The problem statement, all variables and given/known data I have been going through some past exam papers and have come across this vector space question that I cannot find relevant examples for. Consider the vector space V of n-th order polynomials p(x) = a0 + a1x + a2x^2 +· · ·+anx^n, where a0,a1,a2, ...,an are real numbers, and n is a fixed positive integer. Show that the vector space V is closed under addition, and also under multiplication with a real scalar. Also, what is the dimension of V, and determine a set of basis functions for V. Determine an inner product for V. 3. The attempt at a solution I have looked through a couple of relevant textbooks that covered the theory vaguely but did not show how a question such as this should be approached. Any assistance with this question would be appreciated. Is the dimension simply n+1? Also, is the basis simply an?