Linear Algebra Difficulty: Introductory

Systems of Linear Equations

Solving linear systems is a core task in scientific computing. Gaussian elimination and matrix factorizations provide efficient and numerically stable methods.

Key Points

  • A system $A\mathbf{x} = \mathbf{b}$ has unique solutions when $A$ is invertible.
  • Gaussian elimination with partial pivoting improves numerical stability.
  • LU decomposition factorizes $A$ into lower and upper triangular matrices.

Formulas

Linear system
$$A\mathbf{x} = \mathbf{b}$$
LU factorization
$$A = LU$$
Condition number
$$\kappa(A) = \|A\| \|A^{-1}\|$$

Code Example

import numpy as np

A = np.array([[2, 1], [1, 3]])
b = np.array([4, 5])
x = np.linalg.solve(A, b)
print(x)

Tags

  • systems
  • gaussian-elimination

References

  • Introduction to Linear Algebra
    Gilbert Strang · Wellesley-Cambridge Press · source
  • Numerical Linear Algebra
    Lloyd N. Trefethen and David Bau III · SIAM · source

Knowledge Graph