Given a basis [itex]\mathfrak{B}=\lbrace\mathbf{e}_{i}\rbrace[/itex] it is possible to represent a vector [itex]\mathbf{v}[/itex] as a column vector(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\left[\mathbf{v}\right]_{\mathfrak{B}}= \left(\begin{matrix}v^{1} \\ v^{2} \\ \vdots \\ v^{n}\end{matrix}\right)[/itex]

where the [itex]v_{i}[/itex] are the components of [itex]\mathbf{v}[/itex] relative to the basis [itex]\mathfrak{B}[/itex].

I understand that the components, [itex]v_{i}[/itex] correspond to the projection of [itex]\mathbf{v}[/itex] onto each basis vector, [itex]\mathbf{e}_{i}[/itex], but is it correct to say that we can consider them as coordinates of [itex]\mathbf{v}[/itex] relative to [itex]\mathfrak{B}[/itex] (where the [itex]\mathbf{e}_{i}[/itex] define the coordinate axes of the given basis), due to the fact that it is always possible to represent any set of basis vectors in the basis that they define, such that the resulting column vectors will `look like' the standard basis?

Or can one only talk about 'coordinates' of a vector relative to a given basis if that basis is 'ordered'?

Sorry if this is wildly wrong, just starting to get the 'hang' (a bit) of the concept of abstract vector spaces, but still struggling to move away from the specific case of "physical" vectors in Euclidean space.

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# Vector components and its coordinate description in a given basis

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