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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.