- #1
guhan
- 43
- 1
I was revising linear algebra and came across the topic of 'constructing complementary subspace given a subspace' - and since the proof (that used Zorn's lemma) of its (complementary subspace's) existence was not constructive, the author defined an equivalence relation in constructing a complementary (to the subspace W) subspace, U, of the vector space V.. The equivalence relation, [tex]\equiv[/tex]w, is defined by:
(u [tex]\equiv[/tex]w v) iff (u-v) [tex]\in[/tex] W
I understand this algebraically, but I am not able to 'visualize' it correctly. Can some one, please, clarify this with an example... like, let, V=R4 and W={(x1,x2,0,0) | xi [tex]\in[/tex] R}.
How does the space of V get partitioned?
How does a partition of V help in the construction of U?
Is U unique? Because, I can construct only one U for the above V and W. Am I missing something?
(u [tex]\equiv[/tex]w v) iff (u-v) [tex]\in[/tex] W
I understand this algebraically, but I am not able to 'visualize' it correctly. Can some one, please, clarify this with an example... like, let, V=R4 and W={(x1,x2,0,0) | xi [tex]\in[/tex] R}.
How does the space of V get partitioned?
How does a partition of V help in the construction of U?
Is U unique? Because, I can construct only one U for the above V and W. Am I missing something?