MULTILINEAR ALGEBRA Meaning and
Definition
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Multilinear algebra is a branch of mathematics that deals with the study of multilinear mappings and tensors. It provides a framework for understanding and manipulating objects that involve multiple variables simultaneously.
In multilinear algebra, a multilinear mapping is a function that takes multiple inputs and returns a single output, where linearity is preserved with respect to each variable independently. These mappings generalize the notion of linear transformations, which only involve one variable. Multilinear maps have various applications in different fields, including physics, computer science, and engineering.
Tensors are the primary objects of study in multilinear algebra. They are mathematical constructs that represent multilinear maps between vector spaces. Tensors can be thought of as generalizations of vectors and matrices, as they can have multiple indices. The study of tensors involves operations such as addition, multiplication, and contraction, which give rise to properties like symmetry, skew-symmetry, and trace.
Multilinear algebra provides an abstract and rigorous framework for dealing with these objects, allowing for a deeper understanding of their properties and relationships. It encompasses various concepts and techniques, including tensor products, exterior algebras, and symmetric algebras, which are used to describe and manipulate multilinear maps and tensors.
Overall, multilinear algebra is an important branch of mathematics that provides tools for analyzing and manipulating objects involving multiple variables simultaneously. It has a wide range of applications across various fields, and its study contributes to a deeper understanding of the underlying mathematical structures and their interactions.
Etymology of MULTILINEAR ALGEBRA
The word "multilinear algebra" is composed of two components: "multi-" and "linear algebra".
The prefix "multi-" comes from the Latin word "multus", meaning "many" or "much". It indicates that something has multiple or many parts or aspects.
"Linear algebra" is a branch of mathematics that deals with vector spaces and linear transformations between them. The term "linear" refers to the fact that it primarily deals with straight lines or linear relationships between variables.
When these two components are combined, "multilinear algebra" refers to a generalization of linear algebra that allows for multiple linear relationships or multilinear relationships between variables. It focuses on vectors, tensors, and their transformations in a setting where linearity may not hold.
