How to find the orthogonal complement

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Orthogonal complement

Orthogonal complement is defined as subspace $M^\perp = \{ v\in V\,|\, \langle v, m\rangle = 0,\forall m\in M\}$. This is really a subspace because of linearity of scalar product

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6.2: Orthogonal Complements

Finding the orthogonal complement of a span? Let V = P3(R) the vector space of all polynomials in t of degree at most 3. W = M2 × 2(R) the vector space of all 2 × 2 real matrices.

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Orthogonal Complements

Span{e1, e2} ⊥ = {(x y z w) in R|(x y z w) ⋅ (1 0 0 0) = 0 and (x y z w)(0 1 0 0) = 0} = {(0 0 z w) in R4} = Span{e3, e4}: the orthogonal complement of the xy -plane is the zw -plane.

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