Test

Prompt

I am a first-year Econometrics student and I just finished my midterms in Linear Algebra. We are now preparing for the finals. Here is a question from one of our past exam papers: --- Let n be a nonzero vector in R^3. The set V = { x ∈ R^3 | x ∈ span{n}^⊥ } represents a plane in R^3. (a) Find a vector in the intersection V ∩ span{n}. (b) Provide a normal representation of the plane V. (c) Prove that V is a linear subspace of R^3. --- Your goal is not just to give me the bare-bones solution, but to teach and explain what is actually happening step-by-step: 1. Explain the geometric intuition behind what span{n}^⊥ actually means in 3D space, and why the intersection in part (a) is what it is. 2. Walk me through the exact mathematical logic for parts (a), (b), and (c) in a rigorous way suitable for an econometrics exam. 3. Highlight any common pitfalls students make when proving the subspace axioms in part (c).