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Bases and coord systems

  1. Mar 13, 2005 #1

    In abstract linear algebra you learn that vectorspaces (and in particular I
    want to only consider ordered 2-tuples) have no natural coordinate system,
    but you can introduce coordinate systems and describe vectors relative to
    other vectors.

    However, in physics, they often make the (1, 0) and (0, 1) vectors [2-tuple
    vector objects themselves, *not* coordinates] parallel to the page edges,
    regardless of the coordinate system used.

    My question is: Is this merely convention?

    I mean, it seems arbitrary that (1, 0) and (0, 1) get to be made parallel to
    the page edges, although it is intuitive. So is this just the standard
    intuitive way of interpreting the linear vector space of 2-tuples?

    To clarify with an example, consider the ramp problems in physics.

    There are two bases, the ground basis where we know gravity and the ramp
    basis, where one of the axes is coincident with the ramp slope, and the
    other orthogonal with the ramp slope.

    We want to change the gravity vector to be relative to the ramp basis. The
    ground system, which gravity is initially relative to, is taken to be the
    standard basis. Is this necessary or just done for convenience?
  2. jcsd
  3. Mar 14, 2005 #2

    matt grime

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    Homework Helper

    Yes, it is a convention, and it is in some way the easiest to use, just as base 10 is the easiest for us to write numbers in. You need some reference frame, so why not this one? In particular situations better ones may be used (just as base 2 is sometimes better in which to count), but you can't predict when that'll happen.

    In physics we often choose vectors that are orthogonal so we can resolve forces. Sometimes we choose a better basis so that we can work out properties of linear maps (maps of the plane, in this case, that send straight lines to straight lines and keep the origin fixed).

    Learning to do Change of bases is always tricky, but inthe same way that double entry bookkeeping is tricky, and is probably why it is brushed under the carpet.
  4. Mar 14, 2005 #3


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    I might point out that, in physics, there are NO "given" coordinate systems. The choice of which way the x, y, and z axes point and the length of a unit coordinate is the arbitrary choice of coordinate system (or basis for a vector space).

    That's a main reason why physics "laws" are typically in terms of vectors: If an equation in terms of vectors is true in one coordinate system, then it is true in any coordinate system.
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