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December 13th, 2017, 02:45 AM   #1
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Subspaces.

Hello,
I have a doubt. If two subspaces of a vector space have the same size, are they the same?
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December 13th, 2017, 04:00 AM   #2
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I assume that by "same size" you mean "same dimension".

The answer to you question is no. For example take the vector space to be , the set of ordered triples, (x, y, z) with the usual addition and scalar multiplication. Then subspaces {(x, 0, 0)}, {(0, y, 0)}, and {(0, 0, z)} are different subspaces but all have dimension 1. The subspaces {(x, y, 0)}, {x, 0, )}, and {x, y, 0)} are different subspaces that all have dimension 2.
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December 13th, 2017, 06:38 AM   #3
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Thank you.Yes I mean "same dimension".So two subspaces are equal if they have the same base.Right?
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December 13th, 2017, 06:54 AM   #4
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Basis, yes. Although there is no unique basis for any vector space.
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