Is there a connection between the left null space and the annihilator of a matrix? The left null space of a matrix is a subspace. If it's OK to keep it real, the left subspace, $\mathcal L,$ of an $m \times n$ matrix, $M$, is a subspace of $\mathbb R^m.$ In my (limited) understanding, it can be thought of as the (sub)-space orthogonal to the column space, so that the dot product of the vectors in the left null space with the column vectors in $M$ is zero.
On the other hand, the Wikipedia definition of annihilator as it refers to a subset is a follows:

Let $S$ be a subset of $V$. The annihilator of $S$ in $V^*$, denoted here $S^o$, is the collection of linear functionals $f\in V^*$ such that $[f,s]=0$ for all $s\in S.$ 

Linear functionals in $V^*$ can be thought of as row vectors dotted with a elements in a vector space. In this sense, the parallelism with the left null space seems warranted.
So the question is:
Can the annihilator be compared in some way to the left null space?
 A: If you think of the left null space of $M$ as consisting of row vectors, and you also think of the elements of the annihilator of a subspace as being row vectors, then the left null space of $M$ is the annihilator of the range of $M$.
Here is a proof. Let $z$ be a column vector in $\mathbb R^m$. Then
\begin{align}
z^T \in R(M)^\circ &\iff z^T Mx = 0 \quad \text{for all } x \in \mathbb R^n\\
&\iff (M^T z)^T x  = 0 \quad \text{for all } x \in \mathbb R^n\\
&\iff M^T z = 0 \\
&\iff z^T \text{ is in the left null space of $M$.}
\end{align}

Here is a generalization of the above fact.
Let $V$ and $W$ be finite dimensional vector spaces over a field $F$, and let $T:V \to W$ be a linear transformation.  Let $V^*$ and $W^*$ be the dual spaces of $V$ and $W$ (respectively) and let $T^*:W^* \to V^*$ be the dual of $T$, defined by
$$
\langle T^*z, x \rangle = \langle z, Tx \rangle.
$$
Then the annihilator of the range of $T$ is the null space of $T^*$:
$$
R(T)^\circ = N(T^*).
$$
This is a generalization of the "four subspaces theorem" emphasized in Gilbert Strang's linear algebra books. (This theorem is sometimes called the "fundamental theorem of linear algebra".)
Here's a proof:
\begin{align}
z \in R(T)^\circ &\iff \langle z, Tx \rangle = 0 \quad \text{for all } x \in V\\
&\iff \langle T^* z, x \rangle = 0 \quad \text{for all } x \in V \\
&\iff T^*z = 0 \\
&\iff z \in N(T^*).
\end{align}
