Transpose

We believe it helps to first see the transpose of a natrix, and the see the transpose of a vector.

$\begin{bmatrix} 1 & 2\\3 & 4 \end{bmatrix}^T = \begin{bmatrix} 1 & 3\\2 & 4 \end{bmatrix}$

$\begin{bmatrix}4 & 5\end{bmatrix}^T=\begin{bmatrix}4\\5\end{bmatrix}$

One more example:

$A = \begin{bmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{bmatrix}$

$A^T = \begin{bmatrix} a & e & i & m \\ b & f & j & n \\ c & g & k & o \\ d & h & l & p \end{bmatrix}$

Notice that the elements in the diagonal don’t move.