# Math Insight

### The transpose of a matrix

The transpose of a matrix is simply a flipped version of the original matrix. We can transpose a matrix by switching its rows with its columns. We denote the transpose of matrix $A$ by $A^T$. For example, if \begin{align*} A=\left[ \begin{array}{rrr} 1&2&3\\4&5&6 \end{array} \right] \end{align*} then the transpose of $A$ is \begin{align*} A^T=\left[ \begin{array}{rr} 1&4\\2&5\\3&6 \end{array} \right]. \end{align*}

We can take a transpose of a vector as a special case. Since an $n$-dimensional vector $\vc{x}$ is represented by an $n \times 1$ column matrix, \begin{align*} \vc{x} = \left[ \begin{array}{c} x_1\\ x_2\\ x_3\\ \vdots\\ x_n \end{array} \right], \end{align*} the transpose $\vc{x}^T$ is a $1 \times n$ row matrix \begin{align*} \vc{x}^T = \left[ \begin{array}{ccccc} x_1& x_2& x_3& \cdots& x_n \end{array} \right]. \end{align*}