Proving the Direct Sum Decomposition of a Vector Space

[ashina14]

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.

Homework Equations

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?

 

[hedipaldi]

Prove that if v is a nonzero vector in the intersection of U and W, then the u'is must be dependent.

 

[ashina14]

How can I show v is non zero?

 

[Mark44]

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.

 

[ashina14]

Thanks for the help guys :)