Orthogonal complements of the four fundamental subspaces

The four fundamental subspaces come in two perpendicular pairs. One pair lives in the input space and the other in the output space. Keeping those two spaces straight helps us understand both the dimensions and the orthogonality.

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Pair the row space with the null space

If $Ax=0$, each row of $A$ has dot product zero with $x$. So the null space is the orthogonal complement of the row space, which is the column space of $A^T$.

Pair the column space with the left null space

If $A^{T}y=0$, each column of $A$ has dot product zero with $y$. So the null space of $A^T$, called the left null space of $A$, is the orthogonal complement of the column space of $A$.

For an m-by-n matrix of rank r

Perpendicular pair	Ambient dimension	Their dimensions
Row space and null space	$n$	$r$ and $n-r$
Column space and left null space	$m$	$r$ and $m-r$

Work through the dimensions for a four-by-three matrix

$A=\left[\begin{array}{rrr}1& -1& 0\\ -2& 0& 0\\ 0& 3& -1\\ 1& 1& -2\end{array}\right]$

Row reduction gives three pivots, so the rank is $3$. The row space has dimension $3$ and the null space has dimension $0$. The column space has dimension $3$, and its orthogonal complement has dimension $4-3=1$.

A vector spanning that complement is $(-5,-2,-2,1)$. Its dot product with each column of $A$ is zero, which checks that it lies in the left null space.

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