> [!summary] Row space is the transpose of the column space > **Key Equations:** > $\text{Row}(A) = \text{Col}(A^T) = \text{Span}(\left\{\vec{r_{1}},\dots,\vec{r_{n}} \right\} )$ >[!info]+ Read Time **⏱ 2 mins** # Definition The row space of a [[Matrix Notation|matrix]] $A$ is the [[Transposing Matrices|transpose]] of the [[Column Space|column space]]. Since the column space of a matrix $A$ is a [[Subspace|subspace]] of $\mathbb{R}^m$, the [[Transposing Matrices|transpose]] would be a [[Subspace|subspace]] of $\mathbb{R}^n$. ## Derivation > [!warning] Assumptions To derive an expression for the column space, assume the following: > - A [[Linear Mapping (Transformations)|linear transformation]] is from $\mathbb{R}^n\to \mathbb{R}^m$ > - The [[Matrix Mapping (Transformations)|matrix mapping]] is defined as $A \vec{x}=\vec{b}$ > - [[Matrix-Vector Multiplication|Matrix vector multiplication]] can be written as $A \vec{x}= x_{1} \vec{a_{1}}+\dots+ x_{n} \vec{a_{n}}$ > - [[Span|Span]] of some [[Linear Combinations|linear combinations]] is written as $\text{Span}(\vec{v_{1}},\dots, \vec{v_{n}})$ > - The [[Transposing Matrices|transposition]] of the column of a matrix is the row of a matrix. $(a_{i}^T=r_{i})$ $ \begin{array}{c} \begin{align*} \text{Col}(A^T) &= \left\{ \vec{b} \space | \space \vec{b} = A^T \vec{x} \in \mathbb{R}^n \right\} \\ &= \left\{ \vec{b} \space |\space \vec{b} = x_{1} \vec{a_{1}}^T+\dots+x_{n}\vec{a_{n}}^T \space x\in \mathbb{R}^n \right\} \\ &= \text{Span}(\left\{\vec{r_{1}},\dots,\vec{r_{n}} \right\} ) \\ \\ \text{Row}(A) &= \text{Col}(A^T) = \text{Span}(\left\{\vec{r_{1}},\dots,\vec{r_{n}} \right\} ) \end{align*} \end{array} $ The proof of subspace is shown in [[Column Space|column space]], and is not proven here. Since the columns space is a [[Subspace|subspace]] of $\mathbb{R}^m$ then the [[Transposing Matrices|transpose]] of the column space would be a [[Subspace|subspace]] of $\mathbb{R}^n$ by the [[Transposing Matrices|properties of transposing matrices]]. --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidian-MathMatter) with a star and help others learn more easily. ---