Vector Spaces
Cheat sheet.

The key rules, formulas and reminders from all 3 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Span, Independence and Bases

Basis = spanning + linearly independent. A basis gives unique coordinates; its size is the dimension.

Independence belongs to a list, span is a set, and dimension is not necessarily the number of ambient coordinates.

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

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

For $A\in\mathbb R^{m\times n}$ of rank $r$ : column/row dimensions $r$ , nullity $n-r$ , left nullity $m-r$ .

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

2 reference blocks

Basis = spanning + linearly independent. A basis gives unique coordinates; its size is the dimension.

Independence belongs to a list, span is a set, and dimension is not necessarily the number of ambient coordinates.

2 reference blocks

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

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

2 reference blocks

For $A\in\mathbb R^{m\times n}$ of rank $r$ : column/row dimensions $r$ , nullity $n-r$ , left nullity $m-r$ .

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