What is the intersection of nullspaces of S1, S2, and S3?

[mathomatt]

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.

Any advice would be appreciated.

 

[HallsofIvy]

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.

 

[mathomatt]

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