A question in complementing into other basis

[transgalactic]

http://img297.imageshack.us/my.php?image=img86351hl4.jpg

i know that i should add independent vectors
but i don't have any clue about how meny vectors to add
ans which ones
?

 

[HallsofIvy]

I'm not sure what you mean by "complement" the base. I suspect you mean just to "extend" the basis of the smaller space to a basis for the larger. If that is the case you would just add independent vectors that are still the larger space. But that seems awfully trivial! You have, correctly, that a basis for $U\cupV$ is {(1, -1, 0, 0)} and you were told that a basis for U is {(1, 2, 0, 1), (1, -1, 0, 0}. You need to add a vector to {(1, -1, 0, 0} to give a basis for U? Duh!

Extending the basis of U to a basis of $U\cup V$ is similarly trivial. LOOK at the bases you already have!

I am wondering if you are not talking about orthogonal complement. That would mean you need to add a vector, in the larger space, that is orthogonal (perpendicular) to the vectors already in the basis.

 

[transgalactic]

no its not orthogonal complement
the answer in the book is that they just add the missing vector
(the vectors which lack the small groop)
and makes the equal

for example in the example that you presented they add (1,2,0,1)
is that ok?