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Is there a book (e.g. linear algebra book, or maybe geometry book) that explains in great details the geometric meaning of matrices and the effect of operations on them?

I’m not simply talking about a book with a few illustrations for what a rotation does to the plane.

Here is the kind of questions I wish the book answered:

For instance, determinant is area multiplier.

Another example: any matrix can be seen as a rotation, scaling then another rotation. This is what SVD decomposition tells us.

But what is the transpose of a matrix? What’s the geometric effect? Once again SVD, tells us that this is inverse rotations but same scaling. Any other way to understand or “see” this ?

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The following might not be exactly what you're looking for but I personally find Chapter 2: Geometry of Linear Maps, from J. Callahan: Advanced Calculus. A Geometric View, Springer 2010, a very nice intro to the topic on just 42 pages.

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