Tensor space theory I: over a field - Lec 08 - Frederic P Schuller

The video begins with a review of vector spaces, emphasizing the foundational role of fields in their construction. It explains that a vector space is an algebraic structure with a field, denoted as K, which is equipped with two operations, addition and multiplication, satisfying specific properties. The distinction between a physicist's and mathematician's definition of a field is highlighted. The video also introduces the concept of a ring, which is similar to a field but with different properties, and notes the importance of these structures in later discussions on modules.

The script delves into the properties of fields and rings, using examples from high school mathematics to illustrate these concepts. It explains that while every field is a ring, not every ring is a field. The video uses the example of negative integers to show a commutative ring without multiplicative inverses. It also touches upon matrix operations over real numbers to demonstrate a non-commutative ring. The importance of these structures in creating vector spaces is reiterated.

The video introduces the concept of a vector subspace, explaining that it is a subset of a vector space that is closed under the operations of addition and scalar multiplication. The properties that define a subspace are discussed. Linear maps, or homomorphisms, are then defined as structure-preserving maps between vector spaces. The script outlines the properties of linear maps, including additivity and scalar multiplication, and introduces the concept of a vector space isomorphism.

The video explores operations that can be performed on vector spaces, such as defining the set of all linear maps between two vector spaces, which can itself be made into a vector space. The script discusses the addition and scalar multiplication operations on this new space, called the dual space, and how it can be constructed. The importance of understanding the dual space in the context of linear algebra is highlighted.

The concept of tensor products is introduced, explaining how they can be used to construct new tensors from existing ones. The script defines a tensor product as a multilinear map that takes multiple copies of a vector space and its dual, and maps them to a scalar. The properties of tensor products and their role in creating higher-rank tensors are discussed.

The video discusses isomorphisms in the context of vector spaces, highlighting that the dual of the dual of a vector space is isomorphic to the original space, but only if the space is finite-dimensional. The script also touches upon the concept of tensor spaces and their isomorphism to endomorphisms on the dual space. The importance of considering the dimension of a vector space when working with isomorphisms and dual spaces is emphasized.

The script explains the concept of dimension in vector spaces, defining it as the cardinality of a basis set. It distinguishes between finite and infinite-dimensional spaces and discusses the role of a basis in generating a vector space. The properties of a basis, including linear independence and the ability to express any vector in the space as a linear combination of basis vectors, are covered in detail.

The video addresses the process of changing bases in a vector space and the impact it has on the components of vectors and tensors. It explains how the components of a tensor transform under a change of basis and introduces the concept of tensor density. The script also discusses the use of matrices to represent the change of basis and the resulting transformation of tensor components.

The script delves into the concept of an induced basis, particularly in the context of dual vector spaces. It explains how choosing a basis for a vector space leads to an induced basis for its dual space, satisfying certain conditions. The importance of this induced basis in simplifying calculations and the role of the dual basis in defining tensor components are highlighted.

The video discusses how to represent tensors using matrices once a basis has been chosen. It explains the relationship between the components of a tensor, the chosen basis, and how these components can be used to reconstruct the tensor. The script also touches upon the representation of endomorphisms as matrices and the implications of matrix multiplication in terms of endomorphism composition.

The script provides transformation rules for tensors under a change of basis, emphasizing the need to understand where indices are placed and how they transform. It discusses the abstract nature of tensors and the importance of reconstructing tensors from their components. The video also cautions against the temptation to rely on component-based notation, advocating for a basis-free understanding of tensors.

The video concludes with a discussion on determinants, volume forms, and tensor densities. It defines the determinant of an endomorphism and explains its relationship with top forms and volume in a vector space. The script also addresses the concept of tensor densities and how they relate to determinants, providing insight into their role in physics and mathematics.