Mr. Integral
Math notes/Linear Algebra/Vector Spaces

Subspaces, Null Spaces and Column Spaces — Cheat sheet

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Core rule

N(A)={x:Ax=0},C(A)=span⁡(columns of A),Ax=b⇒x=xp+N(A).N(A)=\{x:Ax=0\},\quad C(A)=\operatorname{span}(\text{columns of }A),\quad Ax=b\Rightarrow x=x_p+N(A).

Watch for

Use original pivot columns for a column-space basis. A nonzero right-hand side generally gives an affine set, not a subspace.