Given an operator $A$ that acts on [real vector space](Real%20vector%20spaces.md) $\mathcal{V}$, the transpose is the operator $A^T$ that acts on the [dual space](Vector%20spaces.md#Dual%20Spaces) $\mathcal{V}^*$ such that for $\mathbf{u} \in \mathcal{V}^*$ and $\mathbf{v} \in \mathcal{V}$, $(A^T\mathbf{u})\mathbf{v} = \mathbf{u} (A\mathbf{v}).$ # Transpose of a matrix When $A$ can be written as a [matrix](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Matrices) the transpose switches the matrix indices such that $(A^T)_{ij} = (A)_{ji}.$ This means that the columns and rows are switched. #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices