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Linear Algebra - Finding a Basis

  1. Sep 29, 2011 #1
    1. The problem statement, all variables and given/known data
    I am having trouble finding a basis in a given vector space.

    I understand how to find a basis of Rn, just find linearly independent vectors that span Rn

    But how would i find a basis of the set of 3x3 symmetric real matrices?
    Or Find a basis of real polynomials of degree less than or equal to 3?

    If i understand more about a basis, then I might be able to do this.

    2. Relevant equations

    A set of vectors B in a vector space S is a basis of S iff B is a linearly independent spanning set of S.

    3. The attempt at a solution
     
  2. jcsd
  3. Sep 29, 2011 #2

    CompuChip

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    ... which is just fancy talk for (basically): if you take any element of S, you can express it as a linear combination of the elements of B.

    For example, a basis for the space of all 2 x 2 matrices would be
    [tex]\left\{
    B_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},
    B_2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},
    B_3 = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},
    B_4 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
    \right\}
    [/tex]
    because you can write any matrix
    [tex]A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}[/tex]
    as a linear combination, namely:
    [tex]A = a B_1 + b B_2 + c B_3 + d B_4[/tex]
     
    Last edited: Sep 30, 2011
  4. Sep 29, 2011 #3
    thank you, that helps
     
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