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Omar91X

  • 2 years ago

Let A be a fixed vector in R^(nxn) and let S be the set of all matrices that commute with A; that is, S={B | AB=BA} Show that S is a subspace of R^(nxn).

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  1. satellite73
    • 2 years ago
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    what do you need to show in order to show something is a subspace of a vector space?

  2. Omar91X
    • 2 years ago
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    All I know is that the subset has to satisfy two conditions. That is what I am not sure how to start.

  3. satellite73
    • 2 years ago
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    i think you only need to show two things 1) if \(w, v\in S\) then \(w+v\in S\) and 2) if \(w \in S, \lambda\in \mathbb{R}\) then \(\lambda w\in S\)

  4. satellite73
    • 2 years ago
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    i guess i should have written it with capital letters, but that is the idea

  5. satellite73
    • 2 years ago
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    so you have two jobs 1) show that if \(B, C\) commute with \(A\), that is if \(AB=BA\) and \(AC=CA\) then \[A(B+C)=(B+C)A\] i.e. show that if \(B\in S\) and \(C\in S\) then \(B+C\in S\)

  6. satellite73
    • 2 years ago
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    this should be straight forward because of the distributive law

  7. satellite73
    • 2 years ago
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    you also have to show if \(AB=BA\) then \(A\lambda B=\lambda BA\) which again should be straight forward by the definition of scalar multiplication

  8. Omar91X
    • 2 years ago
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    This answers every question I had. Thanks for taking your time. And then I just realized how simple this should've been.

  9. satellite73
    • 2 years ago
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    yw

  10. satellite73
    • 2 years ago
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    goal of course is to take the general definition and apply it in the specific case that is the hard part, rest is routine

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