Comment by defen | Hacker News Reader

Comment by defen | Hacker News Reader

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I'll take a crack at being as pedantic as possible. I'm just doing this to play along.

1) "Linear algebra is the math of vectors and matrices" - I think this would be better phrased as "the math of vectors and linear transformations." Matrices are a convenient way to represent finite dimensional linear transformations, but I think it's putting the cart before the horse to say that linear algebra is the math of matrices. This is a minor problem, though.

2) "A vector ⃗v ∈ R^n is an array of n real numbers" - In general vectors are not arrays of numbers, real or otherwise. This kind of definition will run into problems when you get into Hilbert/Banach spaces.

3) "the function f(x) = ln(x) has the inverse f^−1(x) = e^x" - this is only true for x > 0 if we're talking about the real-valued logarithm function, since it's not defined for x <= 0.

Thank you, that is awesome. So are all the problems ones of a pedantic nature? For example I understood the inverse function description as a definition even though there was a range (which I also knew) for which it was invalid.

Generally I consider those sorts of things to be the difference between writing 'proofs' versus explaining concepts.

Yeah, I wouldn't really consider any of those things to be "serious" problems, at least for the level that this guide is pitched at.

> 2) "A vector v⃗ ∈ R^n is an array of n real numbers" - In general vectors are not arrays of numbers, real or otherwise. This kind of definition will run into problems when you get into Hilbert/Banach spaces.

Surely this is ok if it is parsed as "(a vector in R^n) is an array of n real numbers".

That's better, but still not entirely correct. An array of real numbers is not itself a vector. It is a representation of a vector in a particular basis. This is a technical point but very important to grok, especially when starting to look at vector spaces other than F^n.

> An array of real numbers is not itself a vector. It is a representation of a vector in a particular basis.

No, such an array can be either a vector or the representation (which also has to be a vector) you mentioned.

To make such things make sense, need a definition of a vector space. It can be tempting to omit that definition from an outline, but the definition is just crucial, e.g., when talking about subspaces, which is often crucial. E.g., for m x n matrix A and n x 1 'vector' x, the set of all x such that Ax = 0 is a subspace, but to know this need the definition of a vector space.

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