Column Space

Definition

$\mathbf{C}(A)={\sum }_{i=1}^n{x}_i{A}_i$ where $A=[A_1,A_2,\dots ,A_n]$

The span of the column vectors of the matrix

Facts

$A\mathbf{x}=\mathbf{b}$ is solvable $\iff$ $\mathbf{b}$ is in the $\mathbf{C}(A)$

$\exists A_{n\times n}^{-1}\iff \mathbf{C}(A)={\mathbb{R}}^n$

non-singular $n\times n$ matrix: $\mathbf{C}(A_{n\times n})={\mathbb{R}}^n$

Column Rank

Definition

$\dim (\mathbf{C}(A))=\text{rank}(A)$

A dimension of the Column Space

Link to original

$\mathbf{C}(A)=\mathbf{C}(A^{⊺}A)$