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: Vectorspace of matrices

  1. Aug 23, 2005 #1
    Hey, I have a quick question. :confused:

    Let [itex]{\cal U}[/itex] be the subspace of [itex]\mathbb{R}_{2x2}[/itex] of all matrices of the form [tex]\left( \begin{array}{ccc}x&-x\\y&z\end{array}\right)[/tex].

    Is it true, that

    [tex]\left( \begin{array}{ccc}x&-x\\y&z\end{array}\right) = x\left( \begin{array}{ccc}1&-1\\0&0\end{array}\right) + y\left( \begin{array}{ccc}0&0\\1&0\end{array}\right) + z\left( \begin{array}{ccc}0&0\\0&1\end{array}\right) = x{\cal M}_1 + y{\cal M}_2 + z{\cal M}_3[/tex]

    So [itex]{\cal B}=\left\{\cal M}_1,{\cal M}_2,{\cal M}_{3} \right\}[/itex] forms a basis for [itex]{\cal U}[/itex]. And that it has dimension 3 after showing that they are independent.


    [tex]{\cal W} = \left( \begin{array}{ccc}a&b\\-a&c\end{array}\right)[/tex]

    \left\{\left( \begin{array}{ccc}1&0\\-1&0\end{array}\right),\left( \begin{array}{ccc}0&1\\0&0\end{array}\right),\left( \begin{array}{ccc}0&0\\0&1\end{array}\right)\right\}

    is a basis for [itex]{\cal W}[/itex] (with three dimensions).

    Any help appreciated. Thanks :smile:
  2. jcsd
  3. Aug 23, 2005 #2


    User Avatar
    Science Advisor
    Homework Helper

    Yeah, you just need to show that M1,M2 and M3 are independent.

    The wording of question 2 is wrong, but I take it's an analogous question which has an analogous solution.
  4. Aug 23, 2005 #3
    Yeah sorry, was slack on that bit. What about the intersection? I get some wacky answer. With a basis; the matrices [tex]\left( \begin{array}{ccc}0&0\\0&1\end{array}\right)[/tex] and [tex]\left( \begin{array}{ccc}1&-1\\-1&0\end{array}\right)[/tex]
    Last edited: Aug 23, 2005
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook