[Linear Algebra] Dimension, Rank

Dimension of Subspace

• dimension of $H$ : number of vectors in any bases of subspace $H$

Column Space of Matrix

• Definition : the column space of a matrix A is the subspace spanned by the columns of A

e.g.

$A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ 0 & 1 \end{bmatrix}$

$\textrm{Col} A = \textrm{Span}\begin{Bmatrix} \begin{bmatrix} 1\\ 1\\ 0\end{bmatrix}, \begin{bmatrix} 1\\ 0\\ 1\end{bmatrix}\end{Bmatrix}$

Matrix with Linearly Dependent Columns

e.g.

$A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix}$

$\textrm{Col} A = \textrm{Span}\begin{Bmatrix} \begin{bmatrix} 1\\ 1\\ 0\end{bmatrix}, \begin{bmatrix} 1\\ 0\\ 1\end{bmatrix}, \begin{bmatrix} 2\\ 1\\ 1\end{bmatrix}\end{Bmatrix} = \textrm{Span}\begin{Bmatrix} \begin{bmatrix} 1\\ 1\\ 0\end{bmatrix}, \begin{bmatrix} 1\\ 0\\ 1\end{bmatrix}\end{Bmatrix}$

Rank of Matrix

• Definition : the rank of a matrix $A$ is dimension of the column space of $A$
• $\textrm{rank} A = \textrm{dim Col} A$