For $A\in\mathbb R^{m\times n}$ of rank $r$, the four fundamental subspaces are: column space in $\mathbb R^m$ (dimension $r$); row space in $\mathbb R^n$ (dimension $r$); right null space in $\mathbb R^n$ (dimension $n-r$); left null space $\ker A^T$ in $\mathbb R^m$ (dimension $m-r$). Orthogonal complement pairs connect row/right-null and column/left-null spaces.
Parents
Children
- Definition of matrix null space (right)The right null space is $\ker A=\{x\in\mathbb F^n:Ax=0\}$ for $A\in\mathbb F^{m\times n}$. It is a subspace of the...
- Definition of matrix null space (left)For real $A\in\mathbb R^{m\times n}$, the left null space is $\ker A^T=\{y:y^TA=0\}\subseteq\mathbb R^m$. For complex...
- Definition of column space of a matrixThe column space $\operatorname{Col}(A)$ is the span of the columns of $A$, equivalently $\{Ax:x\in\mathbb F^n\}$. It...
- The column space of an m-by-n matrix is a subspace of R^m (or C^m)For $A\in\mathbb F^{m\times n}$, $\operatorname{Col}(A)$ is a subspace of $\mathbb F^m$. Indeed, if $u=Ax$ and...
- The pivot columns of a matrix are a basis for the column space.The columns of the original matrix whose indices are pivot columns in an echelon form form a basis of its column...
- Row operations do not necessarily preserve the column space.Row operations need not preserve column space. For $A=(1,0)^T$, swapping its two rows yields $(0,1)^T$, changing the...
- Nonzero rows of reduced row-echelon form give a basis for the row spaceThe nonzero rows of RREF$(A)$ form a basis for the row space of $A$. Elementary row operations preserve row span, and...
- Definition of row space of a matrixThe row space is the span over $\mathbb F$ of the rows of a matrix. Its dimension equals column rank. Over $\mathbb...
- Row equivalent matrices have the same row space.Every elementary row operation replaces rows by invertible linear combinations of rows. Thus the new row space is...
No children match.
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.