The need for the left null space

linear algebramatricesmatrix-rankvector-spacesvectors

I understand intuitive meaning and practical need for column space, row space and null space.

But I can't understand real need for the left null space.

I know that left null space is orthogonal to row space and that it complements full picture of 4 fundamental subspaces.

But what is the practical need/intuition for the left null space, if we know that every vector not in row space have no solutions?

Best Answer

For intuition, a given matrix $A\in\Bbb R^{n\times m}$ determines two linear maps by left or right multiplication:

  1. One taking $n$ dimensional column vectors to $m$ dimensional column vectors:
    $L_A:=\ v\mapsto A\cdot v$
  2. And the other one takes $m$ dimensional row vectors to $n$ dimensional row vectors:
    $R_A:=\ w\mapsto w\cdot A$

The column space is the image of $L_A$, the row space is the image of $R_A$.
The null space is the kernel of $L_A$, and the left null space is the kernel of $R_A$.

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