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Find a basis of the subspace W:=A

  1. Mar 14, 2009 #1
    1. The problem statement, all variables and given/known data
    find a basis of the subspace W:=A[tex]\in[/tex] M2*2(R) : trace (A)=0 of the vector space M2*2 (R) and hence determine the dimension of W

    2. Relevant equations



    3. The attempt at a solution
    trace(A) denote the the sum of the diagonal elements of the matrix A=aij
    do i need to choose some vectors to form a basis ,hence to determine the dimension ?
     
  2. jcsd
  3. Mar 15, 2009 #2

    CompuChip

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    Homework Helper

    Re: dimension

    Well, the elements of M2x2(R) are 2x2 matrices. Recall that a basis of a vector space is a set of elements e1, ..., en such that any element can be expressed as a linear combination of the ei.

    A general 2x2 matrix looks like
    [tex]A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}[/tex]
    so a basis would be, for example,
    [tex]e_{11} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},
    e_{12} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},
    e_{21} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},
    e_{22} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
    [/tex].

    (Aside question for you: how do you express A as a linear combination of these four elements?)
     
  4. Mar 15, 2009 #3
    Re: dimension

    I suggest you write the condition trace(A)=0 in terms of the four matrix elements. Then try to find the simplest examples of matrices which satisfy trace(A)=0 (matrices with only one or two nonzero elements), then try to construct a basis.
     
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