# Intersection of nullspaces

I am looking to find a vector which does not lie in various subspaces.

For example, if I have:
S1 = [1,0,0; 0,1,0] (x-y plane)
S2 = [1,0,0; 0,0,1] (x-z plane)
S3 = [0,1,0; 0,0,1] (y-z plane)

I want to find a vector which was not within any of these subspaces - in this specific example any point that is not on the planes mentioned above. So the point [1,1,1] would be fine.

I am not just wanting to check whether a point is in any of these subspaces, but rather to find a method which will provide me with a point that is definitely not in these subspaces.

I feel that the space I am interested in is the intersection of the nullspaces of S1, S2 and S3, however I am unsure how to find such a space.

HallsofIvy
Homework Helper
I'm not at all sure what you MEAN by "the nullspaces of S1, S2 and S3". Linear transformations have "null spaces", subspaces do not.

Apologies.

Hopefully this makes more sense.

I need to find a vector that cannot be formed from a linear combination of the rows in S1. It also can't be formed by a linear combination of rows in S2 or a linear combination of rows in S3.

So I am trying to find a vector b such that there is no solution to each of the following systems - no such x1, x2 or x3 vectors exist.

S1^T*x1 = b
S2^T*x2 = b
S3^T*x3 = b