Page 1 of 1

GR9768 Q 28

Posted: Tue Apr 06, 2010 3:18 pm
by mathQ
If V1 and V2 are 6 dimensional subspaces of a 10 dimensional vector space V, what is the smallest possible dimension that V1 intersection V2 can have ?

A) 0
B) 1
C) 2
D) 4
E) 6

Re: GR9768 Q 28

Posted: Tue Apr 06, 2010 3:24 pm
by EugeneKudashev
take two 6-dimensional vectors and look at their intersection. it is plain that it contains no less than 2 components.

Re: GR9768 Q 28

Posted: Tue Apr 06, 2010 3:40 pm
by mathQ
EugeneKudashev wrote:take two 6-dimensional vectors and look at their intersection. it is plain that it contains no less than 2 components.
How did you figured out so quickly ? Any tricks ?

which vectors you took ?
thanks

Re: GR9768 Q 28

Posted: Tue Apr 06, 2010 3:46 pm
by EugeneKudashev
perhaps I'll sketch it like that:

| | | | | | x x x x
x x x x | | | | | |
1 2 3 4 5 6 7 8 9 10

first row is your first subspace ( | stands for element, x for emptiness) and the second row is your second subspace. both have dim=6 and as you can see their intersection should not "exceed" dim=10. thus, intersection has dim no less than 2. I hope that is understandable.

Re: GR9768 Q 28

Posted: Tue Apr 06, 2010 3:59 pm
by mathQ
EugeneKudashev wrote:perhaps I'll sketch it like that:

| | | | | | x x x x
x x x x | | | | | |
1 2 3 4 5 6 7 8 9 10

first row is your first subspace ( | stands for element, x for emptiness) and the second row is your second subspace. both have dim=6 and as you can see their intersection should not "exceed" dim=10. thus, intersection has dim no less than 2. I hope that is understandable.
indeed..u have just explained in the right and quick manner. Awesome again!!!
Thanks

Btw, where are you from Eugene ?

Re: GR9768 Q 28

Posted: Tue Apr 06, 2010 4:12 pm
by EugeneKudashev
you're welcome!
I'm from Moscow, Russia, will graduate from Moscow State University this June. you?

Re: GR9768 Q 28

Posted: Sat Oct 01, 2011 3:29 am
by Hom
Hi, i was thinking the same way at first but then I came up with another thought. Not sure if it's valid.

Let a 2D plane p1 to be x=0, and the other plane p2 to be x=2. See p1 and p2 are 2D space of 3D space. But there is no intersection between p1 and p2.

Please comment.

Re: GR9768 Q 28

Posted: Sat Oct 01, 2011 3:55 am
by Hom
Hom wrote:Hi, i was thinking the same way at first but then I came up with another thought. Not sure if it's valid.

Let a 2D plane p1 to be x=0, and the other plane p2 to be x=2. See p1 and p2 are 2D space of 3D space. But there is no intersection between p1 and p2.

Please comment.
Sorry, I forgot the vector space has to contain the origin. x=2 is not a valid 2D space then. p1 and p2 containing the origin will have intersection

Re: GR9768 Q 28

Posted: Wed Oct 19, 2011 10:49 am
by cauchy2012
We have a formula in the textbook of linear algebra: dim(V1)+dim(V2)=dim(V1+V2)+dim(V1intersectV2),so 6+6-10=2 :D

Re: GR9768 Q 28

Posted: Wed Oct 19, 2011 3:31 pm
by goombayao
You can also notice that the kernel of the intersection is at most 8 dimensional. Thus the image must be at least 2 dimensional.