Systems of Linear Equations

Numerical Analysis · Axiom Academy

INTRO Systems of Linear Equations Where do lines meet? Discover how to find the intersection of multiple constraints. Drag the sliders to move the two lines. Can you find where they meet? Click on each system type to see how we write it compactly using matrices. Click each case to visualize what can happen when solving a system: Given Ax = b , find the vector x . For small systems, we can solve by hand. But what about 1000 equations with 1000 unknowns? Gaussian Elimination, LU Decomposition: Systematically transform A into simpler forms to find x exactly (up to round-off). Jacobi, Gauss-Seidel, SOR: Start with a guess, then improve it step by step until we're close enough to the answer. Conditioning matters: Some systems are very sensitive to small errors. Understanding when to trust our answers is crucial.

This is the written version of the interactive lesson above. See the full Numerical Analysis course.