Orthogonality and Projection_什么是orthocomplement of subspace

Orthogonality of the four subspaces

• The row space is perpendicular to the nullspace
• The column space is perpendicular to the nullspace of $A^T$
  
  
• The column space C(A) , a subspace of $R^m$, dimention r
• The leftnull space N($A^T$) , a subspace of $R^m$ ,dimention m-r
• The row space C($A^T$) , a subspace of $R^n$ ,dimention r
• The null space N(A) , a subspace of $R^n$ ,dimention n-r
  
  

Definition of orthogonal subspace

• Two subspace $V$ and $W$ of a vector space are orthogonal if every vector $v$ in $V$ is perpendicular to every vector $w$ in $W$

• Every vector x in the nullspace is perpendicular to every row of $A$, because $Ax=0$ ,the nullspace $N(A)$ and row space C ( A T ) C(A^T)