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 z3_solve.py prove

2. Linear Independence

   • Set up Ax = 0 where columns are vectors
   • sympy_compute.py nullspace "A"
   • Trivial nullspace = independent

3. Basis and Dimension

   • Find spanning set, remove dependent vectors
   • sympy_compute.py rref "A" to find pivot columns
   • Dimension = number of pivots

4. Change of Basis

   • Find transition matrix P
   • New coords = P^(-1) * old coords
   • sympy_compute.py inverse "P"