There are a lot of texts which talks about coordinate and components of a vector and Coordinate system vs basis of vector space, which are all using the terms interchangeably. Nowhere have I found these terms being explained so as to properly differentiate them.

What are coordinates of a vector and components of a vector? I know that the in contravariant vectors, the coordinates of a vector and components of the vector vary the same way and that in a covariant vector they they change oppositely . Also, I have read that when a basis is chosen for the vector space the coordinates of the space (so coordinate system?) are set but that the components of the vector don't need a basis,i.e. components are part of the vector itself and so don't need a basis( I don't understand how components don't need any basis). As per my understanding coordinates of a vector are the coefficients which along with basis vector (i.e. a linear combination of them) give any vector in the vector space(span the vector space). So the coefficients i.e the coordinates of the vector are given relative to the basis. But coordinate of a vector space is (is this coordinate system?) is a function (which is set on choosing the basis set), which maps a vector $V$ (in vector space) to its coordinates (in that basis set). $$x^i[f]v=v^i[f]$$ Where $x^i$ is the coordinate function(does a coordinate system for a vector space mean this coordinate function?) in the basis $f$ and $v$ is the vector in a finite-dimensional vector space and $v^i$ is the coordinates of that vector in basis $f$. So is this the function $x^i[f]$ (that is called the coordinate system of vector space) that vary similar to coordinate(components) of the vector in a contravariant vector? I have also read that coordinates are the function that map region of the euclidian space to points from a manifold but let's not use that explanation. Is the components of the covariant vector mean that they are the dot product of the vector and basis vector,i.e the projection of the vector along each of the basis vectors( if this is so then components of vector do need the basis for their definition too) and that is why they c0-vary with changes to basis in covariant vector?

Also, what is the difference between the coordinate system of a vector space and basis vector for a vector space or more precisely does a change of coordinate system mean the same thing as the change of basis vector set?. I know if the change of basis set is given by a linear map then it is same a change of coordinate system too but is it the only instance where they both are considered similar as the change of coordinate system to a cylindrical or spherical coordinate system won't be considered as a change of basis. What are the definitions of change of coordinate system and of change of basis and is the change of coordinate matrix the same as the change of basis matrix?


1 Answer 1


This is not a proper answer, but it might be useful anyway. It's just what I've found so far reading books and trying to make sense of everything I learned about vectors in both calculus and algebra.

You can geometrically picture vectors in $\mathbb{R}^n$ as arrows placed at the origin. Every vector can be uniquely expressed as a linear combination of $n$ linearly independent vectors, so for every basis of this vector space $(\vec{e}_1,\dots,\vec{e}_n)$ we can write: $$ v=v^1 \vec{e}_1 + \cdots + v^n \vec{e}_n $$ Although using a vector basis allows us to uniquely identify every vector in $\mathbb{R}^n$, it isn't really useful when trying to identify every "point" in $\mathbb{R}^n$ because of its limitations (your axes will have to be straight lines and will have to include the "canonical" origin).

In order to identify every point of $\mathbb{R}^n$ with more freedom, our first approach could be affine geometry. In $\mathbb{R}^n$ viewed as an affine space, we can define a coordinate system $(O,\mathcal{B})$, with $O$ a point in $\mathbb{R}^n$ and $\mathcal{B}$ a basis of $\mathbb{R}^n$ as a vector space. Axes are still straight lines, but now we can move from the origin to any other point $O$. This affine coordinate system is in a way a coordinate system, and it is definitely not the same as a basis — because $(O,\mathcal{B}) \neq \mathcal{B}$ — but we can do better.

We can define a system of $n$ equations (not necessarily linear, a system of linear equations would bring us back to the affine case) that uniquely identify every point in $\mathbb{R}^n$:

$$ x^i =\Phi^i(q^1,\dots,q^n), \space\space\space i=1,\dots,n $$

This is precisely what we do when we define cylindrical or spherical coordinates: express $x,y,z$ in terms of three new variables ($\rho,\varphi,z$ and $r,\theta,\varphi$ respectively).

This means that the values $(q^1,\dots,q^n)$ will be our new coordinates. Note that, if this system were linear, we would need to require the coefficient matrix to have a non-zero determinant in order for this system to have a (unique) solution. For a general system of equations, by virtue of the implicit function theorem, the analogous condition is: $$ \frac{\partial(x^1,\dots,x^n)}{\partial(q^1,\dots,q^n)} \neq 0 $$ i.e. the Jacobian of the transformation must be non-zero.

This new coordinates $(q^1,\dots,q^n)$ aren't related to any basis, but they induce one for every point $(q^1,\dots,q^n)$ in $\mathbb{R}^n$: the so-called coordinate basis of this coordinate system: $$ \vec{v}_\mu = \frac{\partial\vec{\Phi}}{\partial q^\mu} $$

In this sense, we can see than the components of the position vector at the point $p\in\mathbb{R}^n$ will be different from the coordinates of the point $p$ itself. The components depend on the coordinate basis (or any other basis which you define in terms of that one), while the coordinates of a point depend on the coordinate system itself.

In fact, now we're not talking about $\mathbb{R}^n$ as a vector space anymore, but this space does have a vector space attatched at every point, a vector space we call the tangent space at $p$: $T_p\mathbb{R}^n$. The coordinate basis at every point is the vector basis that our coordinate system induces for the tangent space at that point.

Note that the components of a vector (or a tensor, for that matter) may be called coordinates. I only see this when reading about pure vector spaces, without any sense of geometry, metrics or anything. For example, a matrix $A\in\mathcal{M}_{n\times n}$ which satisfies $\vec{v}_{\mathcal{B}'} = A\vec{v}_{\mathcal{B}}$ might be called a change of coordinates matrix (from the coordinates from $\mathcal{B}$ to the coordinates from $\mathcal{B}'$) or a change of basis matrix (from the basis $\mathcal{B}'$ to the basis $\mathcal{B}$, because it satisfies $\mathcal{B}'A=\mathcal{B}$, if we allow ourselves this abuse of notation).

Nonetheless, I always say components when referring to a vector (or tensor), and coordinates when referring to a point.


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