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Fundamentals of Linear Algebra and Optimization [pdf]

seas.upenn.edu

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Re: Fundamentals of Linear Algebra and Optimization [pdf]

#61

Post by kccqzy »

apricot wrote: post #60
kccqzy wrote: post #37

Earlier quoted context omitted.

For me my test for multivariable calculus text is their treatment of the chain rule. If the book says something like “derivative of a composition is the composition of the derivatives” it’s good. If they instead say something involving a sigma and things like ∂f/∂x ∂x/∂t it’s likely bad.

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: https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0f...

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.

Re: Fundamentals of Linear Algebra and Optimization [pdf]

#62

Post by apricot »

kccqzy wrote: post #61
apricot wrote: post #60

Earlier quoted context omitted.

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 lineari…

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.

Re: Fundamentals of Linear Algebra and Optimization [pdf]

#63

Post by begriffs »

wolfgke wrote: post #50
begriffs wrote: post #20

Earlier quoted context omitted.

I think the book you're talking about is Axler's "Linear Algebra Done Right." I worked through the problems in this book with some friends, and it is a very good presentation. After doing that book I wanted to see what other books were out there, which ones were the good ones, and sort of classify them. I came up with this list: https://begriffs.com/posts/2016-07-24-best-linear-algebra-bo... I classify the books as G…

> After doing that book I wanted to see what other books were out there, which ones were the good ones, and sort of classify them. I came up with this list: https://begriffs.com/posts/2016-07-24-best-linear-algebra-bo... Ironically you put "Jänich, K. (1994). Linear algebra. New York: Springer-Verlag." into the "Theoretical" section: In Germany this book (its German original) is not that well-regarded, since it is co…

You're right, I made a mistake. Just reviewed the table of contents on Amazon and it looks pretty short, and not particularly theoretical. Also poorly reviewed. I don't remember how it got on my list but I'm removing it.
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