The tensor product is a mathematical operation that combines two vector spaces to create a new vector space. It takes two vectors from different spaces and produces a new vector that encapsulates the information from both. This operation is widely used in various fields, including physics and engineering, to describe complex systems. In more formal terms, if you have two vector spaces, V and W, their tensor product, denoted as V ⊗ W, consists of all possible linear combinations of the form v ⊗ w, where v is from V and w is from W. This structure allows for the representation of multi-dimensional data and relationships.