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> 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".

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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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