### 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*}

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