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