linear algrebra question (1 Viewer)

[Sirius Black]

Consider the vector space R³. Find four vectors a, a',b,b' such that A=Sp{a,a'} B=Sp{b,b'} and A+B=R³ and AnB(A intesects B)=Sp{(1,1,1)}

does anyone know how to make two planes into a space i.e. R³...

 

[Affinity]

I think sirius overlooked the fact that the + here does not mean the usual union.

it means (direct) sum, which is, all vectors of the form a+b where a is in A and b is in B, because of this A+B will contain more points/vectors than A union B.

by the way, you can also think of A+B as the 'smallest' linear space which contains both of them(formal definition: the intersection of all spaces containing both A and B) this way of thinking is quite useful. when you don't need to construct the elements explicitly

 

[Affinity]

oops slipped there.

yes.. direct means there's only one way to write the sum