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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!

    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.

    Is that basically how these problems go? Thanks.
    Last edited: Feb 20, 2006
  2. jcsd
  3. Feb 21, 2006 #2


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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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