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## Main Question or Discussion Point

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]

(u [tex]\equiv[/tex]

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=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?

_{w}, is defined by:(u [tex]\equiv[/tex]

_{w}v) iff (u-v) [tex]\in[/tex] WI understand this algebraically, but I am not able to 'visualize' it correctly. Can some one, please, clarify this with an example... like, let, V=R

^{4}and W={(x_{1},x_{2},0,0) | x_{i}[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?