A hyperplane is a subspace of one dimension less than the space it is embedded in. In other words, a hyperplane is a flat surface that divides a higher-dimensional space into two or more distinct regions.

For example, in a two-dimensional space, a line can be thought of as a hyperplane as it divides the space into two distinct regions – one on each side of the line. Similarly, in a three-dimensional space, a plane can be considered a hyperplane as it divides the space into two distinct regions – one on each side of the plane.

Another example of a hyperplane is in a high-dimensional space, such as a space with five dimensions. In this case, a hyperplane would be a four-dimensional flat surface that divides the space into two or more distinct regions.

Hyperplanes are useful in various mathematical and computational applications, such as in machine learning algorithms where they are used to classify data points into different categories. For instance, in a classification problem with two classes, a hyperplane is used to separate the data points belonging to each class by creating a boundary between them. The data points on one side of the hyperplane are classified as belonging to one class, while the data points on the other side are classified as belonging to the other class.

In summary, a hyperplane is a flat surface that divides a higher-dimensional space into two or more distinct regions. It can be thought of as a line in a two-dimensional space, a plane in a three-dimensional space, and a higher-dimensional flat surface in a space with more than three dimensions. Hyperplanes are useful in various mathematical and computational applications, such as in machine learning algorithms for classification.