What is it called when a row matrix is "flipped" into a column matrix?

Transposition.

Even if a row matrix and a column matrix are the same dimension and contain the same elements, they are considered different types. A "T" superscript is used when a column matrix is written as a row of numbers. The first element of a column matrix is the topmost (corresponding to the leftmost element when written as a row.)

The picture shows two different ways of writing out
the *same column matrix*.
To change an element, enter a new value into its box and __hit enter__.
When the column matrix is written in a row the
superscript "T" shows that it is really a column matrix.
The elements of the column matrix have the same subscripts
no matter how the column matrix is displayed.

A "T" superscript on column matrix with means to flip the column into a row, resulting in a row matrix.