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zbowman75 Group Title

Let V be the set of 2x2 matrices with standard definitions for vector addition and scalar multiplication. Determine whether V is a vector space. If V is not a vector space, show that at least one of the 10 axioms does not hold. Have to show that the set of all real symmetric matrices, that is, the set of all matrices such that A^t=A is a vector space.

  • one year ago
  • one year ago

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  1. KingGeorge Group Title
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    So which axioms are you having trouble showing?

    • one year ago
  2. zbowman75 Group Title
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    All of them. I just do not understand for some reason.

    • one year ago
  3. KingGeorge Group Title
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    I can walk you through some of them, and hopefully you'll be able to finish the rest.

    • one year ago
  4. zbowman75 Group Title
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    ok

    • one year ago
  5. KingGeorge Group Title
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    Oh, before I start, we're assuming this space of symmetric matrices has real valued entries, and has 2x2 matrices. Correct?

    • one year ago
  6. zbowman75 Group Title
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    yes

    • one year ago
  7. KingGeorge Group Title
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    So if \(v\in V\), \(v\) looks like \[\begin{bmatrix}a&b\\b&a\end{bmatrix}\]for \(a,b\in\mathbb{R}\). Axiom 1: If \(u,v\in V\), then \(u+v\in V\). Pf: Let \[u=\begin{bmatrix}a&b\\b&a\end{bmatrix},\quad v=\begin{bmatrix}c&d\\d&c\end{bmatrix}.\]Then \[u+v=\begin{bmatrix}a+c&b+d\\b+d&a+c\end{bmatrix}.\]This is still a symmetric matrix, so axiom 1 is satisfied. Make sense?

    • one year ago
  8. zbowman75 Group Title
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    yes.

    • one year ago
  9. zbowman75 Group Title
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    So then Axiom 2 would be satisfied right?

    • one year ago
  10. KingGeorge Group Title
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    Almost. There's one more small thing we have to show. Axiom 2: If \(u,v\in V\), then \(u+v=v+u\). Pf: We know\[u+v=\begin{bmatrix}a+c&b+d\\b+d&a+c\end{bmatrix}=\begin{bmatrix}c+a&d+b\\d+b&c+a\end{bmatrix}=v+u.\]So axiom 2 is satisfied because addition of real numbers is commutative.

    • one year ago
  11. KingGeorge Group Title
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    Axiom 3: If \(u,v,w\in V\), then \((u+v)+w=u+(v+w)\). Pf: I think you can do this yourself. Just let \(u,v\) be the same matrices as above, and let \[w=\begin{bmatrix}e&f\\f&e\end{bmatrix}\]Then add the matrices together, and use the fact that addition of real numbers is commutative.

    • one year ago
  12. KingGeorge Group Title
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    Axiom 4: There exists \(\vec{0}\in V\) such that \(\vec{0}+v=v+\vec{0}=v\) for all \(v\in V\). Pf: Again, I think you can do this. Let \[\vec{0}=\begin{bmatrix}0&0\\0&0\end{bmatrix}\]and do the necessary operations.

    • one year ago
  13. zbowman75 Group Title
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    yes those last two are starting to make sense now.

    • one year ago
  14. KingGeorge Group Title
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    Sorry, for Axiom 3, I should have said "use the fact that addition of real numbers is associative."

    • one year ago
  15. zbowman75 Group Title
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    I knew what you meant.

    • one year ago
  16. KingGeorge Group Title
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    Axiom 5: For all \(v\in V\) there exists some \(-v\in V\) such that \(v+(-v)=-v+v=\vec{0}\). Pf: Another very straightforward one. \[v=\begin{bmatrix}a&b\\b&a\end{bmatrix},\quad -v=\begin{bmatrix}-a&-b\\-b&-a\end{bmatrix}.\]Add them together.

    • one year ago
  17. KingGeorge Group Title
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    Axiom 6: For all \(c\in\mathbb{R}\) and \(v\in V\), \(cv\in V\). Pf: Let\[v=\begin{bmatrix}a&b\\b&a\end{bmatrix}.\]Then\[cv=c\begin{bmatrix}a&b\\b&a\end{bmatrix}=\begin{bmatrix}ca&cb\\cb&ca\end{bmatrix}.\]This is still in the form we want, so this axiom is satisfied.

    • one year ago
  18. zbowman75 Group Title
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    ok starting to make sense now thanks.

    • one year ago
  19. KingGeorge Group Title
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    I think you'll be able to handle the rest. Just make sure to take things one step at a time. Also, for the last axiom, use \[\vec{1}=\text{Id}_2=\begin{bmatrix}1&0\\0&1\end{bmatrix}\]

    • one year ago
  20. KingGeorge Group Title
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    Call me back if you need help with any more of the axioms.

    • one year ago
  21. zbowman75 Group Title
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    I have another question for you. Let V=\[\left\{ \left(\begin{matrix}\cos t \\ \sin t\end{matrix}\right) \right\}\] and define \[\left(\begin{matrix}\cos t_1 \\ \sin t_1\end{matrix}\right)⊕ \left(\begin{matrix}\cos t_2 \\ \sin t_2\end{matrix}\right)=\left(\begin{matrix}\cos (t_1+t_2) \\ \sin (t_1+t_2)\end{matrix}\right)\] c ⨀\[\left(\begin{matrix}\cos t \\ \sin t\end{matrix}\right)=\left(\begin{matrix}\cos ct \\ \sin ct\end{matrix}\right)\] Show that if ⊕ and ⨀ are the standard componentwise operations, then V is not a vector space. I'm not quite sure it is not a vector space.

    • one year ago
  22. KingGeorge Group Title
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    I think your problem will be with axiom 5: For all \(v\in V\), there exists a \(-v\) such that \(-v+v=v+(-v)=\vec{0}\). There is a \(\vec0\) in this space (try to find this), but you may not have the inverses.

    • one year ago
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