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R^n/Orthogonal complement/Example

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Let , endowed with the standard inner product. For the one-dimensional linear subspace , generated by the standard vector , the orthogonal complement consits of all vectors , where the -th entry is . For a one-dimensional linear subspace , generated by a vector

the orthogonal complement can be found by determining the solution space of the linear equation

The orthogonal space

has dimension ; this is a so-called hyperplane. The vector is called a normal vector for the hyperplane .

For a linear subspace that is given by a basis (or a generating system) , , the orthogonal complement is the solution space of the system of linear equations

where is the matrix formed by the vectors as rows.