Proving the Direct Sum Decomposition of a Vector Space

ashina14
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Homework Statement



Suppose B = {u1, u2.. un} is a basis of V. Let U = {u1, u2...ui} and W = {ui+1, ui+2... un}. Prove that V = U ⊕ W.

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The Attempt at a Solution



I think I should prove that elements in U are not in W and viceversa. Then this prove it is indeed a disjunction?
 
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Prove that if v is a nonzero vector in the intersection of U and W, then the u'is must be dependent.
 
How can I show v is non zero?
 
ashina14 said:
How can I show v is non zero?
You assume that v is nonzero - that's what "if v is nonzero" means. You don't need to show the things that you are assuming.
 
Thanks for the help guys :)
 
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