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Basis for a Matrix

  1. Jun 1, 2013 #1
    I'm having trouble finding the spanning set and basis for the matrix;

    | a b |
    | c d | with condition that b=d

    I'm thinking thinking the spanning set would be
    A= x
    B = y
    C = z

    Such that x,y,z are all reals, but I can't think of how to find a basis for this, I'm thinking of doing row echolon form but am thinking of how to set parameters.
     
  2. jcsd
  3. Jun 1, 2013 #2

    tiny-tim

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    Homework Helper

    Hi Offlinedoctor! :smile:
    I don't understand this at all. :redface:

    The members of the spanning set will all be matrices.

    Try again. :smile:
     
  4. Jun 2, 2013 #3
    Are you looking for a basis for the subspace of all 2x2 matrices such that both entries in the second column are equal ?

    Or are you only dealing with a single particular matrix in which case saying "basis for the matrix" would make no sense. Generally when we refer to a basis with regards to a single matrix we are referring to a basis for its column space, row space, or null spaces, of the columns and rows. In the context of linear algebra a basis is a minimal spanning set for a vector space.

    Add some more detail to statement of the problem.
     
  5. Jun 2, 2013 #4

    HallsofIvy

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    I think I answered this before- on a different forum (and for a different poster user name).

    I suspect you are asking for a subspace of all 2 by 2 matrices of the form
    [tex]\begin{bmatrix} a & b \\ c & d \end{bmatrix}[/tex]
    such that b= d.

    Such a matrix looks like
    [tex]\begin{bmatrix}a & b \\ c & b \end{bmatrix}= \begin{bmatrix}a & 0 \\ 0 & 0 \end{bmatrix}+ \begin{bmatrix}0 & b \\ 0 & b \end{bmatrix}+ \begin{bmatrix}0 & 0 \\ c & 0\end{bmatrix}[/tex]
    [tex]= a\begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix}+ b\begin{bmatrix}0 & 1 \\ 0 & 1\end{bmatrix}+ c\begin{bmatrix}0 & 0 \\ 1 & 0 \end{bmatrix}[/tex]
    If that does not answer your question, you need to talk to your instructor.
     
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