1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: [Algebra] R4/U?

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data
    Well it isn't so much the problem as it is the notation used within the problem. But here is the question:

    Determine whether or not [itex]\overline{w}[/itex] and [itex]\overline{v}[/itex] are linearly independent in R4/U

    2. Relevant equations
    If v [itex]\in[/itex] V then [itex]\overline{v}[/itex] = v + U

    3. The attempt at a solution

    I don't understand the notation R4/U (I understand R4 and the subspace U, but don't understand the slash between them)
  2. jcsd
  3. Feb 5, 2012 #2


    User Avatar
    Science Advisor

    it's called a "quotient space" and its elements are called "cosets" and consist of "parallel translates of a subspace by a vector".

    that's why v has the overline: it's the SET:

    {v+u: u in U}.

    which can be thought of as the entire subspace U, moved in the direction/distance of v.

    another way to think of it, is as regarding the entire space (in this case R4) losing dim(U) dimensions, by regarding all points in U as "equivalent (essentially 0)".

    if dim(U) = 1, each coset is a parallel line, and you need a 3-vector to tell you "which line".

    if dim(U) = 2, each coset is a parallel plane, and you need a 2 vector to tell you "which plane".

    higher dimensions are harder to visualize, but the same sort of logic applies.

    simce v+U is a set, v is just a "representative", and the same coset v+U can have different representatives.

    one common way quotient spaces arise is in analyzing linear maps: often, we don't care about the kernel of a linear map (because everything in it just maps to the 0-vector), so we "mod it out". the resulting quotient space is isomorphic to the image space (this is pretty much equivalent to the rank-nullity theorem, but in a more abstract setting).

    you calculate with elements in R4/U pretty much like you do with elements in R4, but with a "+U" along for the ride:

    v+U + w+U = (v+w)+U
    a(v+U) = av+U

    the overline notation is a bit "cleaner" but hides some of what is going on.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook