Can someone help me understand the instructor’s use of the term (and visual) for a fundamental square or basis? It comes up at 0:19 of this video here.

He takes two vectors and then plots the AREA within them to be a “fundamental square or basis”. Then he transforms the area using the matrix. The transformation makes sense, the use of an area doesn’t. My understanding of a vector is it is not an area, it’s a distance and direction. Sure, you could create a square by defining an area to be that, but where is this happening and what would the formula be for that area? It’s not the dot product or cross-product since those result in a scalar or another vector.

According to the instructor, the ‘fundamental square’ area is created by the vectors

\begin{pmatrix}
1 \\
0
\end{pmatrix}

and

\begin{pmatrix}
0 \\
1
\end{pmatrix}

It may be helpful if you watch the previous video closely.
You will find a preview of how not only the area but also the entire coordinate plane transform. You may consider the area transformation (acc. to your doubt) as the minimal view of this coordinate plane transformation. This and the next khan academy page may help you visualize.

fundamental square area = "the minimal view of this coordinate plane transformation ".
Genius! This make sense and it ties it all together for me. Thank you. I was looking for a formula and now understand it to be a representation to aid in understanding.

Fundamental square area is still the area (a square in this case) represented by the vectors (1, 0) and (0, 1). (It is, in a way, a minimal representation of xy-axis system)

Now, suppose, you go ahead and tranform. The tranformation matrix goes and transforms every points on this square (as well as on the entire coordinate plane). After this transformation, the resultant product may not remain a square anymore (possibly, a parallelogram as given in the previous video). However, it is now the minimal view of the transformation.