Understanding Vector Invariance Under Coordinate Transformation

[marschmellow]

What does it mean for a vector to remain "invariant" under coordinate transformation?

I think I already know the answer to this question in a foggy, intuitive way, but I'd like a really clear explanation, if someone has it. I know all of multivariable calculus and quite a bit of linear algebra, yet I am still not sure exactly what people mean by "coordinates." My understanding is that a vector is a list of numbers. I don't know what it means for a vector to be in a certain coordinate system. It seems like the vector <2, 0, 10, 4> is just that: the ordered list of numbers two, zero, ten, four. There isn't any extra information, like "in rectangular coordinates" or "in these particular skew coordinates."

So when a vector remains invariant under a change of coordinates, or "doesn't care" about which coordinates you use, as some texts have put it, what does that mean? Does that mean the vector <2, 0, 10, 4> remains the vector <2, 0, 10, 4> or does it mean that the exact list of numbers changes, but the drawing of the vector looks the same? This seems to me like a dumb question, and I'm guessing the answer is the second option, but linear algebra is often taught so abstractly and so far removed from actual numbers that it isn't obvious to me.

 

[SteveL27]

I think I already know the answer to this question in a foggy, intuitive way, but I'd like a really clear explanation, if someone has it. I know all of multivariable calculus and quite a bit of linear algebra, yet I am still not sure exactly what people mean by "coordinates." My understanding is that a vector is a list of numbers. I don't know what it means for a vector to be in a certain coordinate system. It seems like the vector <2, 0, 10, 4> is just that: the ordered list of numbers two, zero, ten, four. There isn't any extra information, like "in rectangular coordinates" or "in these particular skew coordinates."

Have you taken linear algebra yet? This is all made clear there.

In any event, <2, 0, 10, 4> represents a particular point (or vector) in Euclidean 4-space, but it's a representation of that point relative to the standard basis. The standard basis consists of the vectors $e_1 = (1,0,0,0), e_2 = (0,1,0,0), e_3 = (0,0,1,0), e_4 = (0,0,0,1)$. I hope you don't mind if I use regular parens '(' and ')' to denote vectors, I think that's the more standard notation as opposed to angle brackets. It's convenient to identify points and vectors, as long as we remember that when we talk about a point, we're really talking about a representative of a vector that is based at the origin.

But forget 4-space, that's too hard to visualize. Just think about the plane. If you have the vector (or point) P = (3,2), you can express P as $3e_1 + 2e_2$ where $e_1$ and $e_1$ are (1,0) and (0,1) respectively. I hope that's clear ... let me know if it's not.

Now, what if we picked two arbitrary vectors $v_1$ and $v_2$ in the plane, as long as they did not lie on the same line (when based at the origin). If you dropped perpendiculars from the point P = (3,2) to each of the lines from the origin through $v_1$ and $v_2$, you could (via the parallelogram law) express (3,2) as a linear combination of $v_1$ and $v_2$; for example,

$P = av_1 + bv_2$

In that case, (a,b) would be the coordinates of P with respect to the basis $v_1$ and $v_2$.

Now, if you've taken linear algebra, then this is hopefully reminding you of something you already know. But if you haven't taken linear algebra, I just threw a lot of information at you ... but if you think about it a bit, perhaps it will make sense.

In any event, a point is a point is a point ... but we can express the coordinates of the same point with respect to different bases, in many different ways.

So when a vector remains invariant under a change of coordinates, or "doesn't care" about which coordinates you use, as some texts have put it, what does that mean? Does that mean the vector <2, 0, 10, 4> remains the vector <2, 0, 10, 4> or does it mean that the exact list of numbers changes, but the drawing of the vector looks the same? This seems to me like a dumb question, and I'm guessing the answer is the second option, but linear algebra is often taught so abstractly and so far removed from actual numbers that it isn't obvious to me.

A given vector is the same vector, regardless of how we identify its coordinates with respect to a particular basis.

The vector is pointing in a certain direction, with a certain length, in space. If you "moved space around the vector" but left the vector alone, its coordinates would change, but it would still be the same vector. That's just another way of looking at expressing a vector in a different basis.

 

[valjok]

You are right. The basis is an extra information, indeed. The relationship between coordinates and the basis (matrix) was exposed in http://en.wikipedia.org/w/index.php?title=Change_of_Basis&oldid=394612249
It was removed though in a later edit to hide the concept and misguide us to unnecessary thinking that a basis transformation is a rotation.

 

[lavinia]

I think I already know the answer to this question in a foggy, intuitive way, but I'd like a really clear explanation, if someone has it. I know all of multivariable calculus and quite a bit of linear algebra, yet I am still not sure exactly what people mean by "coordinates." My understanding is that a vector is a list of numbers. I don't know what it means for a vector to be in a certain coordinate system. It seems like the vector <2, 0, 10, 4> is just that: the ordered list of numbers two, zero, ten, four. There isn't any extra information, like "in rectangular coordinates" or "in these particular skew coordinates."

So when a vector remains invariant under a change of coordinates, or "doesn't care" about which coordinates you use, as some texts have put it, what does that mean? Does that mean the vector <2, 0, 10, 4> remains the vector <2, 0, 10, 4> or does it mean that the exact list of numbers changes, but the drawing of the vector looks the same? This seems to me like a dumb question, and I'm guessing the answer is the second option, but linear algebra is often taught so abstractly and so far removed from actual numbers that it isn't obvious to me.

Here is a picture that I find helpful.

Think of the sphere in 3 space at and each of its points imagine the vectors that are tangent to the sphere at that point. These vectors lie in a plane.They are geometrically determined and do not require any coordinates.

Now write down coordinates on the sphere around this point and write down the corresponding coordinates for the tangent vector at the point. This expression writes the geometric vector in terms of some basis. If you choose a second coordinate system then the same vector will have a representation in the new coordinates as well. But the two representations describe the same vector.

Since each representation writes the vector down in terms of some basis, there is a linear transformation that describes this change of basis. The vector is said to be invariant under this linear transformation since the transformation.

 

[marschmellow]

A given vector is the same vector, regardless of how we identify its coordinates with respect to a particular basis.

The vector is pointing in a certain direction, with a certain length, in space. If you "moved space around the vector" but left the vector alone, its coordinates would change, but it would still be the same vector. That's just another way of looking at expressing a vector in a different basis.

Ah ha, that was the explanation I was looking for. I just finished "Calculus III" in high school, but since it's high school you have a whole year and spend a lot more time in class, so there was enough time to go through all of multi and most of linear algebra, so everything you were saying was perfectly familiar. I was very confused because all along I was equating "coordinates" and "components." The components of a vector (in the standard basis) remain invariant, but the coordinates are weights of each basis vector, which obviously vary under a change of basis--varying either covariantly or contravariantly, a concept that I think I understand now.

The wikipedia article and the tangent vector explanation also helped clear things up; thanks for the responses.

 

[Rasalhague]

The components of a vector are the scalar coefficients of the basis vectors when that vector is expressed as a linear combination of basis vectors, in a particular basis. You could call them the coordinates of the vector. A basis determines a coordinate system for a vector space. In general, the components of a vector vary with a change of basis. The vector itself is thought of as staying the same when a new basis is chosen; only the numbers used to express it change.

If the only kinds of vector space that you're familiar with are those whose vectors are defined as lists of numbers, it might help to look up other examples of vector spaces. In any basis, the components of any kind of vector will be a list of numbers, but not all vectors are themselves, by definition, a list of numbers. As a starting point, check out the vector space axioms, which are what define the concept "vector space".

Equations which do not change at all with the transformation [of coordinates] (that is, the terms of which are invariants) are called invariant. Equations which remain valid because their terms, though not invariant, transform according to identical transformation laws, are called covariant.

- Bergmann: Introduction to the Theory of Relativity.

(Note that covariant has another, quite different meaning, when used in opposition to contravariant.)