> [!summary] The left null space is the transpose of the null space > **Key Equations:** $\mathcal{N} (A) = \left\{A^T \vec{r} \space | \space \vec{r} \in \mathbb{R}^m \right\}$ >[!info]+ Read Time **⏱ 1 min** # Definition The left null space of a [[Matrix Notation|matrix]] $A$ is the [[Transposing Matrices|transpose]] of the [[Null Space|null space]]. So by definition, the left space ($\mathcal{N}$) is defined below. Since the nullspace is a [[Subspace|subspace]] of $\mathbb{R}^n$ the transpose of the null space is a subspace of $\mathbb{R}^m$ by the [[Transposing Matrices|properties of transposing matrices]]. $ \begin{align*} \mathcal{N} (A) &= \text{Null}(A^T) \\ &= \left\{A^T \vec{r} \space | \space \vec{r} \in \mathbb{R}^m \right\} \end{align*} $ --- > 🧪 Think this could help someone else? [Star Math & Matter on Github](https://github.com/rajeevphysics/Obsidian-MathMatter) to help more learners discover it. ---