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Linear transformations first, then matrices as a representation of transformations with respect to a specified basis. Depending on your tastes, you may find it too dry, but imo it is a clean presentation.
Do you have any other tests for calculus, analysis, abstract algebra, discrete mathematics, probability or statistics books? For me my test for multivariable calculus text is their treatment of the chain rule.
What do you mean by "derivative of a composition is the composition of the derivatives"? You need to think of a derivative as just a function that takes a function, a point, and returns a linearized function that best approximates the original function at that point. So basically the chain rule states that if you have two functions F, G composed together and you want to find the derivative the linearized approximation , you simply compute the linearized function for both F and G, and compose the linearized version afterwards.
You can read the Wikipedia article on the chain rule and it eventually gives this simple and elegant formula for the chain rule: But the unfortunate reality is that too many textbooks formulate the chain rule in such a complicated manner that it obscures the simplicity and elegance of the chain rule. Thank you for the explanations.
I agree with you that this more abstract view of the chain rule and the derivative itself is superior to the sum-of-products formula one usually sees in a first course in multivariable calculus, but I feel most students have to learn the complicated, technical version first before they can see the beauty of the more abstract one.
Tomte on Dec 17, Stupid question: I would teach linear algebra with a book that barely mentions linear transformations. I would rather the students to learn deeper mathematics only in matrix notation, rather than to master less substantial mathematics with both notations. Teaching both notations may reinforce the idea that matrices and linear transformations are different mathematical objects in finite dimension.
Teaching the more abstract notation is mainly useful in infinite dimension Hilbert spaces. This is exactly backwards. Whether you're discussing the Jacobian of a function, or change of basis matrices, learning the matrix formula is a lot less useful than seeing how it falls out of the linear function definition.
The formula is hard to memorize and gives no intuition for why anything is true. But from the linear function definition it is easy to reconstruct the formula. In fact this is so true that I would say that anyone who only knows the matrix definition does not actually understand linear algebra.
You have succumbed to the coordinate virus, http: Matrices represent linear transformations with respect to a given basis. Linear transformations are completely independent from any chosen basis. And you would forever cripple your students. As potential users of linear algebra, they would be mentally mutilated beyond all hope of regeneration.
That's easy. Vector spaces and linear transformations are best understood abstractly. I may be a troglodyte, but I was bored senseless in my linear algebra, calculus, discrete mathematics, statistics, etc classes in college. Then they tag all the interesting stuff like AI we didn't call it Machine Learning back in the 90s at the end of your major where you actually use it.
I had to go back and relearn it all because I didn't pay attention the first time.
Note to mathematicians or at least math professors: It'll make learning easier and more fun. As a mathematician I say: What is considered as interesting is different for each person. I, personally, for example deeply love this really abstract stuff but of course I am aware that other people have different preferences. So I would say even finding "interesting stuff" that many students in the lecture hall might be interested in is really, really hard.
Another important argument against your idea is: To be even able to formulate the ideas from AI, one first have to learn and understand the words of the language in which one will formulate this.
And these words are like "vector space", "linear map", "tensor product" etc. Starting with advanced topics, such as your AI example, is like giving beginner language learners a really advanced text in the foreign language that they just begin to learn. In other words: A reallz dubious idea.
Considerung your post I can only ask you why you did not have a talk with the course advisor of your faculty. He would immediately have told you why these lectures are important for the things that you are actually interested in. I never gave a crap about algebra, or calculus, until AP physics. When I learned that the point and the origin story!
Seconding your wish that curricula would lead with motivations and then drill mechanics rather than drilling mechanics for 11 years and finally giving you the motivation in year It reminds me a bit of the motivation behind fast.
I found it easier to learn things like stats, linear algebra, and calculus once I had a personal application to it. Even something that I love in Reinforcement Learning, I would sometimes get sleepy reading pages and pages from the Sutton and Barto book, but as soon as I worked on a coding exercise, I could spend hours actively participate with trying to solve the problem. Eridrus on Dec 14, Folks looking for introductory books in this area may also enjoy this new in-progress book: Boyd and Vandenberghe will have a monopoly in anything optimization related if they keep on Cambridge University Press; Lineare Algebra.
Autoren: Beutelspacher, Albrecht Beutelspacher, A. Albrecht , Since the intersection of arbitrarily many linear sets is again linear, we can de- fine the Lineare Algebra von Albrecht Beutelspacher [Buchtipp 7] You are not allowed to sell copies of the free version.