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

1. Linear Independence

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

1. Basis and Dimension

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

1. Change of Basis

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

Tool Commands

Sympy_Nullspace

uv run python -m runtime.harness scripts/sympy_compute.py nullspace "[[1,2,3],[4,5,6]]"

Sympy_Rref

uv run python -m runtime.harness scripts/sympy_compute.py rref "[[1,2,3],[4,5,6]]"

Z3_Prove

uv run python -m runtime.harness scripts/z3_solve.py prove "subspace_closed"

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.