TENSOR PRODUCT Meaning and
Definition
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The tensor product is a fundamental mathematical operation that describes the construction of a new mathematical structure from two existing structures. Specifically, it combines vector spaces, matrices, or modules to create a higher-dimensional space or structure.
In a more formal definition, given two vector spaces or modules V and W over a field or ring, the tensor product of V and W, denoted by V ⊗ W, is the construction that results in a new vector space or module. This new space consists of all possible linear combinations of "tensors," which are formed by pairing elements from V and W.
To understand this operation, it is essential to note that tensors are not straightforward combinations of vectors or matrices. Instead, they possess multiple components, with each component corresponding to a different combination of base vectors or base elements from the original structures. This gives the tensor product the ability to capture more complex relationships or transformations between the constituent parts of V and W.
The tensor product is widely used in various branches of mathematics and physics, including linear algebra, abstract algebra, differential geometry, and quantum mechanics. It provides a powerful tool for expressing and understanding higher-dimensional structures, transformations, and interactions. Furthermore, the tensor product satisfies several important properties, such as associativity and distributivity, which make it a versatile operation for analyzing and manipulating mathematical objects.
Etymology of TENSOR PRODUCT
The word "tensor" comes from the Latin word "tensus" meaning "stretched" or "taut". It was first introduced by the German mathematician Woldemar Voigt in the 19th century. The term "tensor" was used to describe a mathematical object that generalizes vectors and matrices.
The word "product" is derived from the Latin word "productus" which means "produced" or "created". In mathematics, a product typically refers to an operation that combines two or more objects to create a new object.
Therefore, the word "tensor product" combines "tensor" to represent the generalization of vectors and matrices, and "product" to refer to the operation that combines these objects to form a new mathematical structure.
