Suppose $A$ is an $m\times n$ matrix. Then the left null space is defined as $\lns{A}=\nsp{\transpose{A}}\subseteq\complex{m}$.
The left null space will not feature prominently in the sequel, but we can explain its name and connect it to row operations. Suppose $\vect{y}\in\lns{A}$. Then by Definition LNS, $\transpose{A}\vect{y}=\zerovector$. We can then write \begin{align*} \transpose{\zerovector} &=\transpose{\left(\transpose{A}\vect{y}\right)} &&\knowl{./knowls/definition.LNS.knowl}{\text{Definition LNS}}\\ &=\transpose{\vect{y}}\transpose{\left(\transpose{A}\right)} &&\knowl{./knowls/theorem.MMT.knowl}{\text{Theorem MMT}}\\ &=\transpose{\vect{y}}A &&\knowl{./knowls/theorem.TT.knowl}{\text{Theorem TT}} \end{align*}
The product $\transpose{\vect{y}}A$ can be viewed as the components of $\vect{y}$ acting as the scalars in a linear combination of the rows of $A$. And the result is a “row vector”, $\transpose{\zerovector}$ that is totally zeros. When we apply a sequence of row operations to a matrix, each row of the resulting matrix is some linear combination of the rows. These observations tell us that the vectors in the left null space are scalars that record a sequence of row operations that result in a row of zeros in the row-reduced version of the matrix. We will see this idea more explicitly in the course of proving Theorem FS.