/vector-spaces

name: vector-spaces
description: "Problem-solving strategies for vector spaces in linear algebra"
allowed-tools: [Bash, Read]

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"`

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.