Vector Spaces When to Use Use this skill when working on vector-spaces problems in linear algebra. Decision Tree 1. Check Subspace - Contains zero vector? - Closed under addition? - Closed under scalar multiplication? - Verify with 2. Linear Independence - Set up Ax = 0 where columns are vectors - - Trivial nullspace = independent 3. Basis and Dimension - Find spanning set, remove dependent vectors - to find pivot columns - Dimension = number of pivots 4. Change of Basis - Find transition matrix P - New coords = P^(-1) old coords - Tool Commands Sympy Nullspace Sympy Rref Z3 Prove Cognitive T…