Dimension - Linear Algebra

mattmns

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.

quasar987

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.

matt grime

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.