3

I have been studying linear algebra for quite a while now and I still don't really get what a matrix really is. I have found that there are multiple interpretations for matrices - some of them listed below. What other interpretations are there and how do all these interpretations connect? why do matrices coincidentally (or probably for a reason) represent all of these things?

The way I am thinking of matrices at the moment is that matrices are like a table, much like a spreadsheet in Excel and for some weird reason they also happen to represent a system of linear equations, which comes in handy to solve such systems.

Here are the interpretations I know of:

  • Geometrically: Matrices transform space
  • Algebraically: Matrices represent systems of linear equations
  • As a table: Matrices can store information, for example the adjacency matrix for a graph
timtam
  • 848
  • There's one more: the matrix as a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$. Basically, the matrix is a way to generalize the equation y = ax but where y and x live in different dimensions. – Fomalhaut Feb 14 '21 at 17:32
  • I'm reluctant to flag this question as a duplicate, but this question seems very similar, and it might be of use to you. – Joe Feb 14 '21 at 20:31
  • @Joe This is useful, thank you! – timtam Feb 14 '21 at 21:32
  • Hint: determinant of matrices can give us areas. For example, for a polygon with vertices $(x_1, y_2), ... (x_n, x_n)$ and then the area of it will be: $$\frac{\begin{vmatrix} x_1 & x_2\ y_1 & y_2 \ \notag \end{vmatrix}
    • \begin{vmatrix}

    x_2 & x_3\ y_2 & y_3 \ \notag \end{vmatrix}

    • ... + \begin{vmatrix}

    x_n & x_1\ y_n & y_1 \ \notag \end{vmatrix}}{2}$$

     – João Víctor Melo Feb 14 '21 at 17:00
  • @timtam That's great to hear! Linear algebra is amazing when taught right, but mindbogglingly boring when taught wrong. Fundamentally, it's about transforming space. Every concept has a geometric interpretation. A matrix represents a linear transformation, matrix multiplication corresponds to composing transformations, the inverse corresponds to reversing them, the determinant is a volume scale factor... – Joe Feb 14 '21 at 21:46

1 Answers1

1

I can really recommend the videos by 3Blue1Brown on youtube. I think his playlist "the essence of linear algebra" sums up everything you would like to know:

https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

(Simply watch the 3rd video in the series, if you only care about the matrices)

oli H.
  • 116
  • I have watched those videos - they give you a good idea of matrices as linear transformations and what matrices mean geometrically. But Grant doesn't really talk about any other interpretations, or do you have a specific time stamp where he does? – timtam Feb 14 '21 at 17:35
  • Well, in the 2nd video he does explain 3 different ways (perspectives of physics or computer science student) of thinking about matrices/vectors, but I think the best way to think of them is really just to consider them as linear transformations. – oli H. Feb 14 '21 at 17:45