A question in my textbook asks me to show that the first four Hermite polynomials form a basis of P3. I know how to do the problem, but I don't really understand what is behind the scenes. Why can we use the coefficients of a polynomial as a column vector? I don't really know how to ask the question (that means I'm probably really lost, haha). Also, how do you say P3 (like if I was speaking to someone). What exactly is polynomial space? Can it be visualized? I'm confused.