Orthogonality

We have already learned that two vectors are orthogonal when each and every vector in their sub-space are perpendicular every vector in subspace i.e., . But that is not the complete picture.

Task : Observe your room floor/ceiling and the four walls of the, now suppose one of the wall is the sub-space and the ceiling is another sub-space at the intersection wall and ceiling is the line which is (1-D) and this line is common for both the sub-spaces, so this means that the two walls looks perpendicular but are not orthogonal. So when the vector is common in two or more sub-spaces then it must be a zero vector ( It is perpendicular to itself) So the two planes (2-D) in space are not orthogonal but instead was the case in which there is a plane and the line is passing perpendicular to that plane then they both were orthogonal.

RULE: Orthogonality is possible when dimension + dimension dimension { whole space }

E.x. The number of choices for the fundamental sub-space in to orthogonal are (2,1)/(1,2) or (3,0)/(0,3).