Genuinely curious how anyone could know about matrices and _not_ know their relationship to geometry.
Matrices are a notation normally introduced as part of linear algebra, for solving a system of equations. The book I first learned linear algebra from is "Advanced Engineering Mathematics" which doesn't cover anything geometric in the entire first chapter on linear algebra (which usually takes 1 quarter in college to cover). It mentions linear transformations, but only algebraically. What it does talk about is Gauss-Jordan elimination, augmented matrices, homogeneity, inverses, determinants, eigenvalues, unitary and skew-Hermitian matrices.
Most of that stuff isn't geometric, and it seems like it wouldn't be hard for lots of people to have been exposed to the algebra without learning the geometric interpretations.
Everything going on in neural networks today is built on matrices and matrix operations that don't have a naturally geometric point of view.
Because that's how matrices are introduced first, in high school mathematics?
>There isn't always a geometric relationship.
Not sure about that to be honest. Seems like there is always a yin-yang between algebra and geometry -- two sides of the same coin.
>The book I first learned linear algebra from is "Advanced Engineering Mathematics"
I pulled my copy of the shelf in my office for the first time in 20+ years and took a look:
I see many references to geometry : Example 5. Linear Transformations ... the matrices ... represent a reflection in the line... Exercise 19 (Rotation) Show that the linear transformation ... with matrix... is a counterclockwise rotation ... and so on.
Actually, I realized I'm reading the chapter "Linear Algebra Part II , Matrices..." and you cited "the first chapter on linear algebra, so I read that chapter (Vectors) and I find on page 276 : "Geometric Representation" with a nice figure titled "Geometrical Interpretation of a scalar triple product".
So hmm. Perhaps we're looking at different books? I found that the current (10th) edition is available in Safari Books. Interestingly they have removed some of the geometric references that are present in my 1983 version. However they also included this :
"(d) Computer graphics. To visualize a three-dimensional object with plane faces (e.g., a cube), we may store the position vectors of the vertices with respect to a suitable x1x2x3-coordinate system (and a list of the connecting edges) and then obtain a two-dimensional image on a video screen by projecting the object onto a coordinate plane, for instance, onto the x1x2-plane by setting x3 = 0. To change the appearance of the image, we can impose a linear transformation on the position vectors stored. Show that a diagonal matrix D with main diagonal entries 3, 1, image gives from an x = [xj] the new position vector y = Dx, where y1 = 3x1 (stretch in the x1-direction by a factor 3), y2 = x2 (unchanged), image (contraction in the x3-direction). What effect would a scalar matrix have?
(e) Rotations in space. Explain y = Ax geometrically when A is one o..."
So to be honest I'm not seeing evidence to support your assertion that the geometric interpretation of matrices is absent from that textbook.
Despite your objection, it's a fact that matrices for systems of equations and neural networks both do not have a naturally geometric interpretation. Nor do many of the applications used in my math book: electrical networks, traffic flow, models of commodity markets, etc. You are free to ignore that, but if you do, it might not help you understand why some people learned matrices without a geometric context.
You are looking at a different book than I am. I have 7th edition Kreyszig. I didn't say geometry was absent from the book, I said it was absent from the first quarter/semester of linear algebra in my book. There is geometry in the book, but many students that went through college the same time I did could easily have taken the minimum requirement, one quarter of math and stopped there.
I don't know about yours, but my high school math introduced solving systems of linear equations. It did not cover orthonormal bases or rotations or affine transforms or homogeneous coordinates.