Mr. Integral
Math notes/Linear Algebra/Vector Spaces

Rank, Nullity and the Four Fundamental Spaces — Cheat sheet

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Core rule

For A∈Rm×nA\in\mathbb R^{m\times n} of rank rr: column/row dimensions rr, nullity n−rn-r, left nullity m−rm-r.

Watch for

Keep input and output spaces separate. Solvability requires b⊥N(AT)b\perp N(A^T).