In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines.

Is every hyperplane a subspace?

As you correctly note, hyperplanes are not necessarily "vector subspaces", which can be seen from the fact that they do not contain the zero vector. However, every hyperplane is an affine subspace of Rn.

Is hyperplane a vector space?

What is a Hyperplane? In mathematics, a hyperplane H is a linear subspace of a vector space V such that the basis of H has cardinality one less than the cardinality of the basis for V. In other words, if V is an n-dimensional vector space than H is an (n-1)-dimensional subspace.