A finite dimensional space is a mathematical concept that refers to a vector space with a finite basis. This means that the space is made up of a finite number of vectors that can be used to create any other vector in the space through linear combinations.
While a finite dimensional space has a finite number of basis vectors, an infinite dimensional space has an infinite number of basis vectors. This means that an infinite dimensional space can contain an infinite number of linearly independent vectors, while a finite dimensional space cannot.
Some examples of finite dimensional spaces include physical spaces such as a room, a sphere, or a cube. They can also include mathematical spaces such as the space of polynomials of degree n or the space of n-by-n matrices.
The dimension of a finite dimensional space is the number of basis vectors that make up the space. It is also equal to the number of coordinates needed to uniquely describe any vector in the space.
Finite dimensional spaces are used in a variety of scientific fields, such as physics, engineering, and computer science. They are essential for solving problems involving systems of equations, optimization, and data analysis. They also provide a framework for understanding and modeling real-world phenomena.