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Decomposing space of 2x2 matrices over the reals

  1. Mar 31, 2017 #1
    1. The problem statement, all variables and given/known data
    Consider the subspace $$W:=\Bigl \{ \begin{bmatrix}
    a & b \\
    b & a \end{bmatrix} : a,b \in \mathbb{R}\Bigr \}$$ of $$\mathbb{M}^2(\mathbb{R}). $$

    I have a few questions about how this can be decomposed.

    1) Is there a subspace $$V$$ of $$\mathbb{M}^2(\mathbb{R})$$ such that $$W\oplus V=\mathbb{M}^2(\mathbb{R})$$? If so, what is one?

    2) Further, is there a different (i.e., $$\ne V$$) subspace with the same property? And if not, is there a different proper subspace $$U$$ such that $$W+U= \mathbb{M}^2(\mathbb{R})$$?
    Which examples, if any, would work for these questions? I haven't made much progress, so seeing explicit examples would help.

    2. Relevant equations


    3. The attempt at a solution

    I tried considering this. Set $$V= \{A\in \mathbb{M}^2(\mathbb{R}): A^T =-A\}$$, but it didn't work.
     
  2. jcsd
  3. Mar 31, 2017 #2

    Math_QED

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    Homework Helper

    A good start would be to find the dimension of ##W##. Then, you know that the dimension of ##V##, if it exists, should be ##4 - dim(W)##.
     
    Last edited: Mar 31, 2017
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