What does it mean that vector is independent of coordinate system

[aaaa202]

Hi PF, I have always wondered what was meant when my teachers told me that a vector is the same no matter what coordinate system it is represented in. What is it exactly that is the same? I mean the components change. So the only thing that I can see remains the same is the length of the vector. Unless you understand vector as something more abstract than I do. Please explain :)

 

[mfb]

Your description of the vector can change if you express it in a different basis, even if the vector does not.

For example, consider the vector space of polynomials:

X^2+X+2 is a vector in it.
In the basis (1,X,X^2,...) it can be expressed as (2,1,1,0,0,...)
In the basis (1,(X-1),(X-1)^2,...) it can be expressed as (0,-1,1,0,0,...)

 

[BobG]

Conceptually, it means a little more than just the values of the components and the length.

It also affects how the rotation works when going from one coordinate system to another. It also spills over into other concepts. The cross-product of two vectors works regardless of the coordinate system you're using, the dot product of two vectors, laws of conservation are still applicable regardless of the coordinate system, etc. (sometimes there's an advantage to using one coordinate system over the other and you choose that alternate coordinate system with no penalty).

 

[voko]

The components change, but the length does not. That is invariant number one, scalar product with itself. Scalar products with other vectors are also invariant. Geometrically, that means that the length and direction of the vector are unaffected by changes of the coordinate system. The length and direction are the true "identity" of the vector, not its coordinates.

 

[HallsofIvy]

Another way to think about it is that wind blows in a particular direction at a particular speed. That is its "velocity vector". What coordinate system you use, how you measure angles, even whether you measure speed in "miles per hour", "km per hour", or "meters per second", won't affect the wind at all! It will still blow in the same direction at the same speed. It velocity vector is the same no matter what coordinate system you use.

 

[D H]

Hi PF, I have always wondered what was meant when my teachers told me that a vector is the same no matter what coordinate system it is represented in. What is it exactly that is the same? I mean the components change.

Get a blank sheet of white paper. Draw two dots somewhere on that sheet, label them A and B. Draw a directed straight line segment from point A to point B. That directed line segment is a vector. You didn't need a coordinate system to draw it.

Now imagine putting a transparency sheet with grid lines atop that white sheet of paper. Thanks to that grid you can now read off a numerical representation of that vector. Rotate the transparency by 45 degrees and you'll get a different set of numbers. It's still the same vector. All that has changed is how you are representing it. The thing that the vector represents, the displacement from point A to point B hasn't changed.How you choose to represent a vector and the thing that the vector represents are two different things.