Vector components and its coordinate description in a given basis

["Don't panic!"]

Given a basis $\mathfrak{B}=\lbrace\mathbf{e}_{i}\rbrace$ it is possible to represent a vector $\mathbf{v}$ as a column vector

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

where the $v_{i}$ are the components of $\mathbf{v}$ relative to the basis $\mathfrak{B}$.

I understand that the components, $v_{i}$ correspond to the projection of $\mathbf{v}$ onto each basis vector, $\mathbf{e}_{i}$, but is it correct to say that we can consider them as coordinates of $\mathbf{v}$ relative to $\mathfrak{B}$ (where the $\mathbf{e}_{i}$ 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.

 

[mathman]

What do you mean when you say a basis is "ordered"? The answer to your first question is yes.

 

["Don't panic!"]

What do you mean when you say a basis is "ordered"? The answer to your first question is yes.

Thanks.

By "ordered" I mean that there is a specific order to the elements of a given basis, such that rearranging any of the elements in a given basis results in a new (distinct) basis?!

 

[Mandelbroth]

Thanks.

By "ordered" I mean that there is a specific order to the elements of a given basis, such that rearranging any of the elements in a given basis results in a new (distinct) basis?!

Define $\mathfrak{B}'=\{\mathbf{e}_j'\}$ as a basis, with $\mathbf{e}_i'=\mathbf{e}_{n-i+1}$.

Then, $\mathfrak{B}=\mathfrak{B}'$. I suspect you're thinking of "ordered pairs." As it turns out, we can order them for convenience, but the fact remains that vector spaces are abelian groups under addition, so $\mathbf{e}_1+\mathbf{e}_2=\mathbf{e}_2+\mathbf{e}_1$.

Our notation with vectors in columns and rows acts like a function, sending an element of a field in an entry to a corresponding (predetermined) basis vector scalar-multiplied by that field element. Thus, when we talk of coordinates, we ARE talking about ordered sets.

 

["Don't panic!"]

Define $\mathfrak{B}'=\{\mathbf{e}_j'\}$ as a basis, with $\mathbf{e}_i'=\mathbf{e}_{n-i+1}$.

Then, $\mathfrak{B}=\mathfrak{B}'$. I suspect you're thinking of "ordered pairs." As it turns out, we can order them for convenience, but the fact remains that vector spaces are abelian groups under addition, so $\mathbf{e}_1+\mathbf{e}_2=\mathbf{e}_2+\mathbf{e}_1$.

Our notation with vectors in columns and rows acts like a function, sending an element of a field in an entry to a corresponding (predetermined) basis vector scalar-multiplied by that field element. Thus, when we talk of coordinates, we ARE talking about ordered sets.

So is it correct then to say that the reason why we can consider the components of a vector as coordinates of that vector relative to a given basis because of 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? In this sense, is it correct to say the the elements of a given basis define a coordinate system, with each basis vector defining a particular coordinate axis within that system?

 

[1MileCrash]

An ordered basis is the exact termination my linear algebra book uses, it is just a basis regarded as a sequence rather than a set of linearly dependent, space-generating vectors.

Just throwing that out there. I thought the term was common. I don't see how the concept isn't required, and I think the answer to the OP's question "can one only talk about 'coordinates' of a vector relative to a given basis if that basis is 'ordered'?" is yes.

 

[AlephZero]

I don't see how the concept isn't required, and I think the answer to the OP's question "can one only talk about 'coordinates' of a vector relative to a given basis if that basis is 'ordered'?" is yes.

You can only use vector or matrix notation to do numerical calculations in linear algebra, using some specified ordered basis.

But there is a lot more to the mathematics of linear algebra than "vectors and matrices whose elements are real or complex numbers".