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: Vector spans M2,2

  1. Apr 24, 2009 #1
    Hi,I have a question like this :
    Determine whether the given set of vector spans M2,2
    {(5 3;0 0),(0 0;5 3),(6 -1;0 0),(0 0;6 1)}

    I wonder can I jus directly prove that
    a(5 3;0 0)+b(0 0;5 3)+c(6 -1;0 0)+d(0 0;6 1) = (5a+6c 3a-c;5b+6d 3b+d)
    is also a M2,2 and hence,it is a vector sets spans M2,2?

    Thanks a lot!
  2. jcsd
  3. Apr 25, 2009 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    The definition of a set of vectors spanning the whole of the vectors space is what? Because you've not stated it, nor verified it is true.

    Note that this being M_2,2, is irrelevant: it is just a 4-d vector space, so there's no need to write things as matrices.
  4. Apr 25, 2009 #3


    User Avatar
    Science Advisor

    No, the fact that the particular linear combination is in M2,2 only means that these matrices span some subspace of M2,2- which is true of any collection of matrices in M2,2. What you must prove is that any matrix in M2,2 can be written as a linear combination of them.

    (u v; x y) is in M2,2 for any real numbers u, v, x, y. Can you find a, b, c, d such that a(5 3;0 0)+b(0 0;5 3)+c(6 -1;0 0)+d(0 0;6 1)= (u v; x y) for any u, v, x, y?

    Also, since, as Matt Grime said, this is a 4 dimensional space and your set contains exactly 4 matrices, this set spans M2,2 if and only if it is independent. If a(5 3;0 0)+b(0 0;5 3)+c(6 -1;0 0)+d(0 0;6 1)= (0 0; 0 0), what must a, b, c, d be?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook