linalgkit-lab 0.1.1

Comprehensive linear algebra library for education and research

linalgkit

A comprehensive, general-purpose Linear Algebra library for Python — built directly from a complete university syllabus covering six modules.

pip install linalgkit-lab

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Modules Covered

Module	Topic	Functions
I	Linear Systems & Gaussian Elimination	solve_linear_system, gaussian_elimination, gauss_jordan_elimination, row_echelon_form, reduced_row_echelon_form, elementary_matrix, …
II	Vector Spaces	are_linearly_independent, span_basis, subspace_sum, subspace_intersection, is_direct_sum, extend_to_basis, …
III	Linear Transformations	kernel, image, rank_nullity, matrix_of_transformation, change_of_basis_matrix, is_isomorphism, …
IV	Orthogonality	gram_schmidt, gram_schmidt_qr, orthogonal_complement, is_orthogonal_matrix, is_unitary_matrix, project_onto_subspace, …
V	Eigenvalues & Eigenvectors	eigenvalues, eigenspaces, diagonalize, spectral_decomposition, orthogonal_diagonalize_symmetric, unitary_diagonalize_hermitian, …
VI	Canonical Forms	classify_quadratic_form, classify_conic, classify_quadric, verify_cayley_hamilton, minimal_polynomial, bilinear_form, …

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Quick Examples

import linalgkit as la
import numpy as np

# ── Module I: Solve a linear system ─────────────────────────────
A = [[2, 1, -1], [1, -1, 2], [3, 2, 1]]
b = [8, -2, 11]
result = la.solve_linear_system(A, b)
print(result['solution_str'])
# Unique solution: x = [2. 3. 1.]

# Show row-reduction steps for lab work
worked = la.solve_linear_system(A, b, step=True)
for item in worked["steps"]:
    print(item["operation"])

# Infinite solutions
A2 = [[1, 2, 3], [4, 5, 6]]
b2 = [1, 2]
res2 = la.solve_linear_system(A2, b2)
print("Null space basis:", res2['null_space_basis'])

# ── Module I: Elementary matrices ────────────────────────────────
E = la.elementary_matrix(3, 'swap', 0, 1)           # swap rows 0,1
E2 = la.elementary_matrix(3, 'scale', 0, scalar=2)  # scale row 0
E3 = la.elementary_matrix(3, 'add', 1, j=0, scalar=-3)  # R1 -= 3*R0

# ── Module II: Vector spaces ─────────────────────────────────────
vectors = [[1, 0, 0], [0, 1, 0], [1, 1, 0]]  # third is redundant
basis = la.span_basis(vectors)
print("Basis size:", len(basis))   # 2

U_basis = [[1, 0, 0], [0, 1, 0]]
V_basis = [[0, 0, 1]]
print(la.is_direct_sum(U_basis, V_basis))  # {'is_direct_sum': True, ...}

# ── Module III: Linear Transformations ───────────────────────────
A = [[1, 2, 3], [0, 1, 4], [0, 0, 1]]
k = la.kernel(A)
im = la.image(A)
rn = la.rank_nullity(A)
print(f"Rank={rn['rank']}, Nullity={rn['nullity']}, Domain dim={rn['domain_dim']}")

# Matrix of transformation w.r.t. custom bases
def T(v): return np.array([v[0]+v[1], v[0]-v[1]])
M = la.matrix_of_transformation(T, [[1,0],[0,1]], [[1,0],[0,1]])

# ── Module IV: Orthogonality ─────────────────────────────────────
vectors = [[1, 1, 0], [1, 0, 1], [0, 1, 1]]
orth_basis = la.gram_schmidt(vectors, normalize_output=True)
print("Orthonormal?", la.is_orthonormal_set(orth_basis))  # True

comp = la.orthogonal_complement([[1, 0, 0]], ambient_dim=3)
print("Complement dim:", len(comp))  # 2

# ── Module V: Eigenvalues & Eigenvectors ─────────────────────────
A = [[3, 1], [1, 3]]
spaces = la.eigenspaces(A)
for sp in spaces:
    print(f"λ={sp['eigenvalue']}, AM={sp['algebraic_multiplicity']}, GM={sp['geometric_multiplicity']}")

P, D, Pinv = la.diagonalize(A)
print("Verified:", np.allclose(P @ D @ Pinv, A))  # True

sd = la.spectral_decomposition(A)
print("Spectral recon correct:", np.allclose(sd['verification'], A))

# ── Module VI: Canonical Forms ───────────────────────────────────
A = [[1, 0], [0, -2]]
res = la.classify_quadratic_form(A)
print(res['classification'])   # Indefinite

conic_A = [[1, 0], [0, 2]]
print(la.classify_conic(conic_A)['type'])   # Ellipse

quadric_A = np.diag([1.0, 2.0, 3.0])
print(la.classify_quadric(quadric_A)['type'])   # Ellipsoid

ch = la.verify_cayley_hamilton([[1, 2], [3, 4]])
print("Cayley-Hamilton:", ch['verified'])   # True

Lab Notebook Helpers

Most notebook-style routines are available directly from the package:

la.compute_rank_with_rref(A, step=True)
la.solve_homogeneous_system(A, step=True)
la.check_linear_independence(vectors, step=True)
la.find_basis_and_dimension(vectors, step=True)
la.check_direct_sum(U_basis, W_basis, ambient_dim=3, step=True)
la.verify_rank_nullity_theorem(A, step=True)
la.gram_schmidt_orthogonalization(vectors, step=True)
la.orthogonal_complement_basis(basis, ambient_dim=3, step=True)
la.compute_eigen(A, step=True)
la.diagonalize_matrix(A, step=True)

When step=False (the default), existing return values are preserved. When step=True, functions return a dictionary with the computed result and a steps field containing the intermediate operations or explanation.

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Edge Cases Handled

• Inconsistent systems → detected and reported cleanly
• Rank-deficient matrices → null-space basis always returned
• Dependent vectors → Gram-Schmidt skips, span_basis reduces
• Singular matrices → matrix_inverse_via_gauss_jordan returns None
• Non-diagonalizable matrices → diagonalize returns None
• Non-symmetric input to quadratic form → auto-symmetrized
• Complex matrices → full complex inner product / unitary support

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Requirements

• Python ≥ 3.8
• NumPy ≥ 1.21

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License

MIT