Do hyperplanes form subspaces of Rⁿ?

[samh]

Here's the page: http://mathworld.wolfram.com/Hyperplane.html

They say that the set S is a subspace of R^n. Is that true? Doesn't c have to be zero in order for S (the hyperplane) to be a subspace?

 

[CompuChip]

Let n = 2, a₁ = a₂ = 1. If I rename x₁ = x and x₂ = y, then according to you only
x + y = 0
defines a line (hyperplane) but
x + y = 1
doesn't?

[edit]I see your point. Probably they should write that it is ... isometric (is that the word here? it's just a line through the origin translated by a constant vector) to a genuine co-dimension one subspace[/edit]

 

[matt grime]

Affine is the word you're looking for: things that look like translations of vector subspaces.