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March 10th, 2010, 12:05 AM   #1
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basis and dimension

The set of all nxn matrices having trace equal to zero is a subspace W of Mnxn. Find a basis for W. What is the dimension of W?
The trace is: a11=+a22+a33+....+anxn=0
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March 10th, 2010, 04:12 AM   #2
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Re: basis and dimension

Since the sum of two matrix with vanished trace is also matrix with vanish trace and the product of matrix with vanish trace and the scalar is also matrix with vanish trace the subset W is a subspace. Its basis are the matrix units . The number of the linear independnt matrix units is n-1. Hence . Each linear independant matrix can be basis of W.
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