I'm a student in mechanical engineering and we have algebra classes. The chapter we're currently working on is bilinear algebra. Because of that, we were introduced to many applications involving matrices. I figured out that mathematic had the curious yet very intuitiv chance to be visual. And I must admit that representing a problem using visuals is quite satisfying because, for me, its gives a sense of "motion". And because of that, I wondered the following.
I already know that we can visualize ${3}\times{3}$ matrices with the intersection of 3 planes. I started to wonder if, like vectors using their properties, we could visually represent them as plane intersections. To make the question a bit more precise, I drew the $3\times3$ matrix, M.
\begin{equation*} M = \begin{pmatrix} 1 & 2 & 0 \\ 2 & -9 & 0 \\ -1 & 0 & 1 \end{pmatrix} \end{equation*}
Visualization of M
I thought that maybe, the intersection of these planes can be a way to "visualize" and for bigger matrices, the idea would be the same with higher dimension plane intersection. But I don't know if it is right.
This question is pure interest for mathematics and culture, but I would be glad to know if we can "view" objects like matrices visually.