Rank

• categories: Linear algebra, Definition

the rank of a matrix $A$ is

• The dimension of the vector space (Column Space) generated by its columns
• The dimension of the Row Space
• Number of independent columns
• Size of the basis

Intuition about rank = 1 :

1 & 3 & 8 \\ 1 & 3 & 8 \\ 1 & 3 & 8 \end{bmatrix}=UV^T=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 8 \end{bmatrix}$$ Intuition about rank = 2 : $$\begin{bmatrix} 2 & 1 & 3 \\ 3 & 1 & 4 \\ 5 & 7 & 12 \end{bmatrix}=C_{3\times 2}\times R_{2\times3}=\begin{bmatrix} 2 & 1 \\ 3 & 1 \\ 5 & 7 \end{bmatrix} \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix}$$