T - Gordon State College
... If TA: Rn → Rk and TB: Rk → Rm are linear transformations, then the application of TA followed by TB produces a transformation from Rn to Rm. This transformation is called the composition of TB with TA, and is denoted by TB ◦ TA. Thus, (TB ◦ TA)(x) =TB(TA (x)). ...
... If TA: Rn → Rk and TB: Rk → Rm are linear transformations, then the application of TA followed by TB produces a transformation from Rn to Rm. This transformation is called the composition of TB with TA, and is denoted by TB ◦ TA. Thus, (TB ◦ TA)(x) =TB(TA (x)). ...
Lecture 8 - Universal Enveloping Algebras and Related Concepts, II
... Graded Ideals. If U is graded and I is an ideal, then I isLcalled a graded ideal provided that if x ∈ I, then also each π i (x) ∈ I. That is, I = i I ∩ U i . If U is graded and I is a graded ideal, then U/I is graded. Inherited Filtrations. If π : A → B is a surjection of algebras and A is filtered ...
... Graded Ideals. If U is graded and I is an ideal, then I isLcalled a graded ideal provided that if x ∈ I, then also each π i (x) ∈ I. That is, I = i I ∩ U i . If U is graded and I is a graded ideal, then U/I is graded. Inherited Filtrations. If π : A → B is a surjection of algebras and A is filtered ...
lat04_0803
... 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 8.4 De Moivre’s Theorem; Powers and Roots of ...
... 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 8.4 De Moivre’s Theorem; Powers and Roots of ...
1 Fields and vector spaces
... is induced by an invertible semilinear transformation of the underlying vector spaces. In particular, the collineations of PG n F for n @ 2 are induced by invertible semilinear transformations of the rank- n 1 vector space over F. This theorem will not be proved here, but I make a few commen ...
... is induced by an invertible semilinear transformation of the underlying vector spaces. In particular, the collineations of PG n F for n @ 2 are induced by invertible semilinear transformations of the rank- n 1 vector space over F. This theorem will not be proved here, but I make a few commen ...
Cartesian tensor
In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is through an orthogonal transformation.The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesian coordinate systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional vector space over the field of real numbers that has an inner product.Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. Sometimes general curvilinear coordinates are convenient, as in high-deformation continuum mechanics, or even necessary, as in general relativity. While orthonormal bases may be found for some such coordinate systems (e.g. tangent to spherical coordinates), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a passive transformation, since the coordinates are changed and not the physical system.