The tensor product is a mathematical operation that combines two vector spaces to create a new vector space. It allows us to construct higher-dimensional spaces from lower-dimensional ones, enabling the representation of complex relationships between different vectors. For example, if we have two vectors from spaces V and W, their tensor product, denoted as V ⊗ W, consists of all possible combinations of these vectors. In more technical terms, the tensor product captures the idea of multilinearity, meaning it respects the linear structure of the original spaces. This operation is widely used in various fields, including quantum mechanics, machine learning, and differential geometry, where it helps in modeling interactions and transformations between different systems.