Orthogonal complement of a subset or subspace

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

• Well, the complement is a subspace, so you get an element that is orthogonal to every element of U. – zagortenay333 Oct 15 '18 at 15:25
• Not sure I'm following you. – zagortenay333 Oct 15 '18 at 15:40
• Ok, how is the orthogonal complement defined there? I believe that you would end up with the orthogonal complement of the linear span. So, as you write, in your case it would be a line and not a plane. – Christian Oct 15 '18 at 15:58
• The orthogonal complement is defined as the set of all vectors in V that are orthogonal to all vectors of the subset U. Again, the issue is that, as far as I understand, if a line not going through the origin is interpreted as the set of position vectors (from origin to point on line), then no plane through the origin can be orthogonal to any vector of the line! So Axler must interpret the line as position vectors with respect to the offset origin point, but that's kinda weird. – zagortenay333 Oct 15 '18 at 16:25
• I agree to that reasoning – Christian Oct 15 '18 at 16:26

The point raised in the question above is caused by an error by me. Specifically, in the first line after 6.45 in Linear Algebra Done Right (third edition), "line in $$\mathbf{R}^3$$" should be "line in $$\mathbf{R}^3$$ containing the origin". Also, in the second line after 6.45, "plane in $$\mathbf{R}^3$$" should be "plane in $$\mathbf{R}^3$$ containing the origin". Sorry about this error, which will be fixed in the next edition of the book.