December 13th, 2017, 02:45 AM  #1 
Newbie Joined: Dec 2017 From: Cosenza Italy Posts: 4 Thanks: 0  Subspaces.
Hello, I have a doubt. If two subspaces of a vector space have the same size, are they the same? 
December 13th, 2017, 04:00 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,919 Thanks: 785 
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. 
December 13th, 2017, 06:38 AM  #3 
Newbie Joined: Dec 2017 From: Cosenza Italy Posts: 4 Thanks: 0 
Thank you.Yes I mean "same dimension".So two subspaces are equal if they have the same base.Right?

December 13th, 2017, 06:54 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,116 Thanks: 2369 Math Focus: Mainly analysis and algebra 
Basis, yes. Although there is no unique basis for any vector space.


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