In the book "Linear Algebra Done Right" Axler says that given a subset U of a vector space V, the orthogonal complement of U is...
Is it necessary to assume that U is a subspace, not just a subset? The reason this confused me a bit is because in the example immediately following the definition of orthogonal complements he says:
"For example, if U is a line in R3 , then the complement of U is the plane containing the origin that is perpendicular to U."
How is Axler interpreting a line that isn't going through the origin (an affine subset)? How can every vector (presumably position vectors) in a plane be orthogonal to a line not going through the origin? Is he thinking of the line as position vectors with respect to the shifted origin point?
Looking at wolfram, U is assumed to be a subspace in the definition: http://mathworld.wolfram.com/OrthogonalComplement.html