Find the Orthogonal Complement and find the direct sum Solution S = span(A) wher

Question

Find the Orthogonal Complement and find the direct sum

Explanation / Answer

S = span(A) where A = 0 1 -1 1 Now, let B = a b c d such that inner product = 0 => b -c + d = 0 => d = c -b B = (a b c d)' ( writing column vector as complement of row vector for my convenience ) B = (a b c c-b)' = a(1 0 0 0)' + b(0 1 0 -1)' + c(0 0 1 1)' => S_|_ = span( (1 0 0 0)' , (0 1 0 -1) , (0 0 1 1) ) Now , direct sum of two subspaces which are orthognal complements of each other always yields the VECTOR SPACE in our case S + S_|_ = the vector space of all 4 element column vectors or you can say S + S_|_ = span( (0 1 -1 1) , (1 0 0 0)' , (0 1 0 -1) , (0 0 1 1) ) or even S + S_|_ = span( (1 0 0 0) , (0 1 0 0) , (0 0 1 0) , (0 0 0 1) )

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