[tubs]

You explain well so what I never understood is how the Jacobians aren't the first derivatives themselves?

Also if you have happen to have any suggestions for linear algebra for someone who uses it without really understanding it (I can write a measurement function for an EKF from scratch OK, but I don't really understand why the maths does what it does) I would really appreciate it.

[mxwsn]

The Jacobian is first derivatives, but for a function mapping N to M dimensions. It's the first derivative of every output wrt every input, so it will be an N x M matrix.

The gradient is a special case of the Jacobian for functions mapping N to 1 dimension, such as loss functions. The gradient is an N x 1 vector.

[Nevermark]

[EDIT] Updated original comment to include matrix dimensions.

If you want a serious text that goes through the relevant linear algebra and optimization mathematics in depth up front, Neural Network Design, 2nd edition is a good one. [Disclaimer, co-author]. We took great pains to walk through every conceptual and mathematical topic before we apply those concepts to machine learning. We use MATLAB a lot, which may or may not be helpful.

Another potential option is "Linear Algebra and Optimization for Machine Learning", which looks good and also starts out with linear algebra before machine learning. I haven't read it, but the first 2020 edition gets good reviews, and a second 2026 edition just came out, apparently with a fair amount of positive revision. Given the speed of change, that's nice to see.