Explore polynomial spaces from a linear algebra perspective in this comprehensive one-hour lecture. Delve into the four-dimensional space P^3, examining useful bases such as the standard (power) basis and the modified Factorial basis. Discover how translations of corresponding functions yield linear transformations, leading to Taylor bases and an algebraic definition of derivatives. Gain insights into the algebraic nature of fundamental calculus concepts, challenging the necessity of real numbers, limits, or tangent slopes. Investigate various bases, including the standard basis and factorial basis, and learn about basis isomorphism correspondence. Analyze the derivative as a linear transformation and explore change of basis matrices. Engage with practical exercises to reinforce your understanding of these advanced mathematical concepts.