Hyperplane

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. This notion can be used in any general space in which the concept of the dimension of a subspace is defined.

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In different settings, hyperplanes may have different properties. For instance, a hyperplane of an $n$ -dimensional affine space is a flat subset with dimension $n - 1$ [1] and it separates the space into two half spaces. While a hyperplane of an $n$ -dimensional projective space does not have this property.

The difference in dimension between a subspace $S$ and its ambient space $X$ is known as the codimension of $S$ with respect to $X$ . Therefore, a necessary and sufficient condition for $S$ to be a hyperplane in $X$ is for $S$ to have codimension one in $X$ .

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