The Base Change Formula is a mathematical tool used in linear algebra and calculus to convert coordinates or representations of vectors and functions from one basis to another. This is particularly useful when working with different coordinate systems or when simplifying complex problems. The formula allows for the transformation of a vector's representation by multiplying it with a change of basis matrix. In the context of vector spaces, the Base Change Formula helps in understanding how linear transformations behave under different bases. It ensures that the properties of the transformation remain consistent, regardless of the basis used, facilitating easier calculations and deeper insights into the structure of the space.