| # Orthogonal complement of a subspace Let be a subspace of . The orthogonal complement of , denoted , is the subspace of that contains the vectors orthogonal to all the vectors in . If the subspace is described as the range of a matrix: , then the orthogonal complement is the set of vectors orthogonal to the rows of , which is the nullspace of . Example: consider the line in passing through the origin and generated by the vector . This is a subspace of dimension : To find the orthogonal complement we find the set of vectors that are orthogonal to any vector of the form , with arbitrary . This is the same set as the set of vectors orthogonal to itself. So we solve for with : This is equivalent to . This equation characterizes the elements of the orthogonal complement , in the sense that any can be written as for some scalars , where The orthogonal complement is thus the span of the vectors : .