> [!Definition] > Given a [[Matrix|matrix]] $A=(a_{ij})\in \text{Mat}_{mn}$. Then the transpose of $A$, denoted $A^{T}$, is given by $A^{T}=(a_{ji})\in \text{Mat}_{nm}$. In words, the rows of $A$ are the columns of its transpose in the same order. > [!Example] > Let $A=\begin{pmatrix}1 & 3 & 5\\ 0 & 2 & 4 \end{pmatrix}\in \text{Mat}_{23}$ > Then $A^{T}=A=\begin{pmatrix}1 & 0 \\ 3 & 2\\ 5 & 4 \end{pmatrix}\in \text{Mat}_{23}$ # Properties > [!NOTE] Lemma (Transpose of a product) > $(AB)^{T }= B^{T }A^{T}$