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Homework Help: Finding a subspace (possibly intersection of subspace?)

  1. Nov 5, 2011 #1
    1. The problem statement, all variables and given/known data

    Let A be the following 2x2 matrix:

    s 2s
    0 t

    Find a subspace B of M2x2 where M2x2 = A (+) B

    2. Relevant equations

    A ∩ B = {0}

    if u and v are in M2x2, then u + v is in M2x2
    if u is in M2x2, then cu is in M2x2

    3. The attempt at a solution

    Let B be the following 2x2 matrix:

    0 0
    r 0

    Because they are both subspace, they intersect at the zero vector and thus the set {0}, the zero subspace, is a subspace of M2x2. We then have

    M2x2 = A (+) B:

    M2x2 = A + B /\ A ∩ B = {0}
  2. jcsd
  3. Nov 5, 2011 #2


    Staff: Mentor

    This doesn't make sense to me. M2x2 is the vector space of 2x2 matrices. It's not a matrix.

    It also doesn't make sense to add a matrix - A - and a subspace - B.

    What is the exact wording of this problem?
  4. Nov 5, 2011 #3


    Staff: Mentor

    That makes more sense, except that I can't read what's in the parentheses in M2x2(ℝ). In my browser it shows up as an empty box. What symbol is that?

    This stuff, too.
    where {A|s,t in ℝ} ℂ M2x2(ℝ)

    s, t in what?
  5. Nov 5, 2011 #4
    This should make it easier haha

    [PLAIN]http://dl.dropbox.com/u/907375/asd.jpg [Broken]

    In case the image isn't showing either, the symbol that isn't showing is the "R" for real numbers, so s,t in R and M(R)
    Last edited by a moderator: May 5, 2017
  6. Nov 5, 2011 #5


    User Avatar
    Science Advisor

    I take it then that you mean B is a subspace of the space of all two by two matrices with real entries. However, you do NOT mean that A is the "matrix" given. Rather, A is the subspace of all two by two matrices, with real entries, of the form
    [tex]\begin{bmatrix}s & 2s \\ 0 & t\end{bmatrix}[/tex].
    Saying that "[itex]A(+)B= M_{22}(R)[/itex]" means that for any numbers u, x, y, z, there exist numbers a, b, c, d such that
    [tex]\begin{bmatrix}s & 2s \\ 0 & t\end{bmatrix}+ \begin{bmatrix}a & b \\ c & d\end{bmatrix}= \begin{bmatrix}u & x \\ y & z\end{bmatrix}[/tex]

    Of course, then we must have
    [tex]\begin{bmatrix} a & b \\ c & d\end{bmatrix}= \begin{bmatrix}u- s & x- 2s \\ y & z- t\end{bmatrix}[/tex]

    Now, what relations must a, b, c, and d satisfy?
  7. Nov 5, 2011 #6
    Closure under addition and closure under multiplication?
  8. Nov 5, 2011 #7
    I barely see what it's asking...

    Given A, I don't have a problem proving that A is a subspace of M22 -- just show there's closure under addition and multiplication. I can find a basis/span, etc.

    For this question, I'm somewhat lost. Since A + B = M22, then B = M22 - A, I'm assuming B has to follow the closure requirements, but is that it? It seems like I'm just missing something pretty big here...
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