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In "Finite dimensional vector spaces" by Halmos (page 23) I read at some point:

The concept of dual space was defined without any reference to coordinate systems.

Also in an older thread here at math.stackexchange I read that:

Now remember that when a vector space is not given to you as $\mathbb{R}^n$, it doesn't normally have a canonical basis, so we don't have a canonical way to write its elements down via coordinates, and so we can't describe hyperplanes by explicit equations like above. (Or better, we can, but only after choosing coordinates, and this is not canonical.)

What I don't understand is what is meant by coordinates? Given the linear functional $l$ with the mapping $\ell:(x_1,\ldots,x_n) \mapsto a_1 x_1 + \cdots a_n x_n$, what are the coordinates here - the $x_{i}$ or the $a_{i}$? What exactly is meant by the phrase "coordinates" and "canonical" in this context? Could someone help me out with this?

Thanks!

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mannequin
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  • If you have a basis $b_n$, the coordinates (with respect to that basis) of a point $x$ are the values $\alpha_n$ such that $x = \sum_k \alpha_k b_k$. In your example, the underlying space is $\mathbb{R}^n$ with the standard/canonical basis $e_1=(1,0,..,0), e_{2}=(0,1,0,...,0),...,e_n=(0,...,0,1)$ and the point in question is $x=\sum_k x_k e_k$. The coordinates are $(x_1,...,x_n)$. In your particular example, the $a_k$ shows how $l$ behaves on the standard basis, for example, $l(e_i) = a_i$. – copper.hat Jul 31 '16 at 16:33

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Both the $a$'s and the $x$'s are coordinates. The $x$'s are the coordinates of a vector, and then $a$'s are the coordinates of a linear functional.

If you've chosen a basis for a vector space, then any vector can be written uniquely as a linear combination of basis vectors, and the coordinate representation of a vector is the column vector whose entries are the coefficients of the linear combination. For the basis $B$, call this vector $[v]_B$.

Then, the coordinate representation of the linear functional is a row vector. Call this vector $[\ell]_B$. Then $\ell(v)$ can be computed from coordinates via the matrix product of the row vector with the column vector.

If you perform a change of basis from $B$ to $B'$, then all of your coordinate representations change. There is some matrix $T$ so that

$$ [v]_{B'} = T [v]_B $$

and correspondingly the representation of the linear functional has to change too:

$$ [\ell]_{B'} = [\ell]_B T^{-1} $$

That most things you write down depends on what basis you choose, and you have to keep changing the representations of everything if you choose is awkward. It makes it hard to know what things are properties of the vector space you study and what things are simply properties of the basis you happened to choose.

The point of the usual definition of linear functional (i.e. as a linear functional rather than as a row vector that varies with the choice of basis) is that it very clearly does not depend on a choice of basis, and is thus a property of your vector space!