vector-spaces
Problem-solving strategies for vector spaces in linear algebra
When & Why to Use This Skill
This Claude skill provides advanced problem-solving strategies for vector spaces in linear algebra. It leverages computational tools like Sympy and Z3 to automate subspace verification, linear independence testing, basis and dimension calculations, and change of basis transformations, making it an essential tool for students, educators, and researchers in mathematics.
Use Cases
- Subspace Verification: Automatically prove if a subset satisfies closure under addition and scalar multiplication using Z3 theorem proving to ensure mathematical rigor.
- Linear Independence Analysis: Determine if a set of vectors is linearly independent by computing the nullspace of a matrix and identifying trivial solutions.
- Basis and Dimension Calculation: Identify the spanning set and calculate the dimension of a vector space by finding pivot columns through Row Reduced Echelon Form (RREF).
- Coordinate Transformation: Simplify change of basis problems by calculating transition matrices and performing inverse matrix operations to find coordinates in new systems.
| 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
Check Subspace
- Contains zero vector?
- Closed under addition?
- Closed under scalar multiplication?
- Verify with
z3_solve.py prove
Linear Independence
- Set up Ax = 0 where columns are vectors
sympy_compute.py nullspace "A"- Trivial nullspace = independent
Basis and Dimension
- Find spanning set, remove dependent vectors
sympy_compute.py rref "A"to find pivot columns- Dimension = number of pivots
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.