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If f≠0 is linear, then codim(ker f)=1.
Let be a vector space over or , and let be a nonzero linear functional. Then its kernel has codimension one:
The kernel of a nonzero functional is a codimension-one subspace, and translates of such kernels are precisely hyperplanes in real vector spaces; see hyperplanes as level sets.
A vector space over a field is a set with operations (functions) and , and an element , satisfying the following for all and :
Let be a linear operator between vector spaces.
If is bijective (one-to-one and onto), then is a linear isomorphism, and we say that and are isomorphic, written .
Let be a vector space. An affine set is called a hyperplane if it has codimension one.
More precisely: if , it is parallel to a unique subspace (see Ω−Ω characterization). Then is a hyperplane iff .
In real vector spaces, hyperplanes are exactly level sets of nonzero linear functionals; see the level-set characterization.
Let be a real vector space.
Proposition: A subset is a hyperplane if and only if there exist a nonzero linear functional and a scalar such that