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Dimension - Linear Algebra

  1. Feb 20, 2006 #1
    There are these questions in the book that ask us to find the Dimension of a particular space. Do I just find a basis for the space, and then the number of elements in that basis is the dimension for the space? Or is there some trick to finding the dimension? Thanks!


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    For example, the first one the book asks is: Find the dimension of 2x2 matricies. So a basis for 2x2 matricies is the following set:

    [tex]\left\{\left(\begin{array}{cc}1&0\\0&0\end{array}\right), \left(\begin{array}{cc}0&1\\0&0\end{array}\right), \left(\begin{array}{cc}0&0\\1&0\end{array}\right), \left(\begin{array}{cc}0&0\\0&1\end{array}\right)\right\}[/tex]

    And this basis has 4 elements, so the dimension of 2x2 matricies is 4.
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    Is that basically how these problems go? Thanks.
     
    Last edited: Feb 20, 2006
  2. jcsd
  3. Feb 21, 2006 #2

    quasar987

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    Dimension if the number of element in a basis whose elements are linearly independent. So find a basis, check for linear dependancy. If it is lin. dep., trash the "spare" elements of your basis.
     
  4. Feb 21, 2006 #3

    matt grime

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    Point of order: a basis is by definition linearly independent. You cannot 'find a basis then check for linear dependency'. Find a spanning set then find the maximal number of linearly independent elements in it, either by inspection or by turning it into a matrix question and using row reductions to put it in echelon form.
     
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