Sign in Subscribe
5 min read linear-algebra-series

Linear Algebra Series - The Four Fundamental Subspaces Solver

Interactive Solver for The Four Fundamental Subspaces - Column Space, Row Space, Null Space and the Left Null Space
Linear Algebra Series - The Four Fundamental Subspaces Solver
The Four Fundamental Subspaces

The Strategy: [ A | I ]

To find all four subspaces at once, we form the augmented matrix [ A | I ] and perform Gauss-Jordan elimination to reach Reduced Row Echelon Form [ R | E ].

  • Row Space C(Aᵀ): Non-zero rows of R.
  • Column Space C(A): Pivot columns from original A.
  • Null Space N(A): Special solutions from free variables (Solve Rx = 0).
  • Left Null Space N(Aᵀ): Rows of E corresponding to zero rows of R.

Matrix Input

×

The Big Picture

Rⁿ (Inputs) Dim = 3 Row Space Dim = r Null Space Dim = n-r Orthogonal Row Space ⊥ Null Space Rᵐ (Outputs) Dim = 3 Column Space Dim = r Left Null Space Dim = m-r Orthogonal Column Space ⊥ Left Null Space A Aᵀ

Solver runs entirely in browser. Built with Math.js.

Published by:

WassupAI

You might also like...

14
Dec
Linear Algebra Series - Change of Basis Matrix Solver

Linear Algebra Series - Change of Basis Matrix Solver

6 min read
06
Dec
Linear Algebra Series - Special Topics

Linear Algebra Series - Special Topics

7 min read
06
Dec
Linear Algebra Series - Principal Component Analysis (PCA)

Linear Algebra Series - Principal Component Analysis (PCA)

7 min read
06
Dec
Linear Algebra Series - Singular Value Decomposition (SVD)

Linear Algebra Series - Singular Value Decomposition (SVD)

7 min read
06
Dec
Linear Algebra Series - Symmetric Matrices and Quadratic Forms

Linear Algebra Series - Symmetric Matrices and Quadratic Forms

7 min read

Member discussion