Finding the Intersection of Subspaces with Given Spanning Vectors

[Tereno]

How do you find the intersection of subspaces when the subspaces are given by the span of 3 vectors?

For example, U is spanned by { X1 , X2 , X3} and V is spanned by { Y1, Y2, Y3}.

Thanks in advance.

 

[Hurkyl]

I think you just do the obvious thing.

What does it mean for a vector to be an element of U?
What does it mean for a vector to be an element of V?

Then you simply ask when can both of them be true!


If you're not yet inclined to do so, allow me to remind you this is an algebra course, so you should try to be thinking about equations.

 

[Euclid]

The first thing I would do would be to eliminate linearly dependent vectors in {X1,X2,X3} and {Y1,Y2,Y3} if possible.
Once you know the dimension, it may help to think geometrically: A vector space of dimension 1 is a line through the origin (thinking in R^3), a vector space of dimension 2 is a plane through the origin, and a vector space of dimension 3 is all of R^3. What is the intersection of two of these objects? This might give you a hint as to the dimension of the intersection. If you can find enough LI vectors in the intersection, you are done.