Some thought about direct sum and

[td21]

I know that if V is a direct sum of U and W,
then
1. V=U+W
2 there is no intersection between U and W


However, in some books there is an equivalent condition:
3.Every v can be expressed uniquely as u+w


Why's that? Why can we be so sure about the word "unique"? Thanks.

 

[td21]

you can also look up here:http://books.google.co.uk/books?id=...AEwAw#v=onepage&q=direct sum uniquely&f=false

 

[Stephen Tashi]

I know that if V is a direct sum of U and W,
then
1. V=U+W
2 there is no intersection between U and W

You aren't making a precise statement. Are U and W supspaces? What do you mean when you say "there is no intersection"? All subspaces contain the zero vector.