Vector Transpose

It's very important to note that in this lesson we emphasize the column vector.

Vector $\vec{x}=\begin{bmatrix} a_1\\ a_2\\ a_3\\ :\\a_n\end{bmatrix}$ is a column vector.

But vectors can also be represented at row vectors.

Vector $\vec{y}= \begin{bmatrix} a_1&a_2 &a_3& …& a_n \end{bmatrix}$ is a row vector.

If you look closely at both vectors, $\vec{x}$ and $\vec{y}$, you will notice that they have the same elements, only one is a column and the other is a row.

It's as if one vector was actually tilted by $90^{\circ}$.

In the world of Linear Algebra we call this change a transpose. The mathematical symbol of a transpose is $T$ and it's used the following way:

$\begin{bmatrix} a_1\\ a_2\\ a_3\\ :\\a_n\end{bmatrix}^T= \begin{bmatrix} a_1&a_2 &a_3& …& a_n \end{bmatrix}$

Or:

$\begin{bmatrix} a_1&a_2 &a_3& …& a_n \end{bmatrix}^T=\begin{bmatrix} a_1\\ a_2\\ a_3\\ :\\a_n\end{bmatrix}$

Equation 1

In short:

$\large\vec{x}^T=\vec{y}$

or

$\large\vec{y}^T=\vec{x}$

Equation 2