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by belfalas 1466 days ago
I am indeed here posting my original question after first trying the Manga Guide to Linear Algebra and finding it was not what I was looking for. Where I wanted visual explanation they went to textbook definitions, not helpful. A few illustrations in the book I did think were valuable so it wasn't a total loss.
1 comments
wrycoder 1466 days ago
LA is about vectors and rotations and stretches of vectors, which is what happens when you multiply a vector by a matrix. That’s what you will be visualizing.

Try the Kahn videos, then watch the 3B1B videos, which are very visual, but somewhat advanced. Or, watch both of them several times in parallel.

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dragontamer 1466 days ago
This is why I asked "what field are you learning Linear Algebra?".

Elsewhere, I've discovered that this poster is going into image processing, which is likely "signals and systems" linear algebra.

In signals and systems, your vectors can have infinite dimension, and these infinite-dimension vectors Fourier-transform into other infinite dimension vectors under a new basis.

Any field with more than "3 dimension" vectors / matricies is very difficult to visualize geometrically. Trying to do so is counter-productive to the understanding of the field. This geometric interpretation is really useful in graphics programming / 3d animation however.

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Or perhaps a more concrete example... your "visualize the matrix in X dimensions" advice just doesn't cut it if you're dealing with an 8x8 matrix JPEG DCT coefficient matrix (https://en.wikipedia.org/wiki/JPEG#Discrete_cosine_transform), unless you can imagine 8-diemsional space in your brain.

On the other hand, imagining the 8x8 matrix as 64 linearly-independent "Basis" to your 64-dimension discrete signal is... easier. (Well... for a definition of easier at least). And the transform from time domain into Fourier domain is a transformation in basis that contains the same information.

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wrycoder 1466 days ago
I missed your other response!

That one and this one are both quite interesting to me - my focus isn’t signals and systems. Thanks!

I was just thinking that the (3D) vector approach would be a good start along the path to mathematical maturity in linear transformations.

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