Notes
... surjects onto g as a Lie algebra with kernel a Lie bialgebra ideal. Thus, g inherits a quasitriangular Lie bialgebra structure from the double D(b). 3. Hopf algebras. Given G a Lie group, consider C ∞ (G) and U g = (C ∞ (G))∗ . The Lie structure on G gives rise to extra structure on these function s ...
... surjects onto g as a Lie algebra with kernel a Lie bialgebra ideal. Thus, g inherits a quasitriangular Lie bialgebra structure from the double D(b). 3. Hopf algebras. Given G a Lie group, consider C ∞ (G) and U g = (C ∞ (G))∗ . The Lie structure on G gives rise to extra structure on these function s ...
Document
... If S is a subspace of , then dim S dim S n Furthermore, if { x1 ,..., xr } is a basis for S and {xr 1 ,..., xn}is a basis for S , then { x1 ,..., xr , xr 1 ,..., xn} is a basis for n. ...
... If S is a subspace of , then dim S dim S n Furthermore, if { x1 ,..., xr } is a basis for S and {xr 1 ,..., xn}is a basis for S , then { x1 ,..., xr , xr 1 ,..., xn} is a basis for n. ...
2.3 Vector Spaces
... b) Using the standard relation between <2 and points on the plane make a sketch ...
... b) Using the standard relation between <2 and points on the plane make a sketch ...
Finite Dimensional Hilbert Spaces and Linear
... F, of scalars. Any two vectors x, y ∈ X can be added to form x + y ∈ X where the operation “+” of vector addition is associative and commutative. The vector space X must contain an additive identity (the zero vector 0) and, for every vector x, an additive inverse −x. The required properties of vecto ...
... F, of scalars. Any two vectors x, y ∈ X can be added to form x + y ∈ X where the operation “+” of vector addition is associative and commutative. The vector space X must contain an additive identity (the zero vector 0) and, for every vector x, an additive inverse −x. The required properties of vecto ...
Week 1 – Vectors and Matrices
... (v1 + w1 , v2 + w2 ) is the composition of doing the translation (v1 , w1 ) first and then doing the translation (v2 , w2 ) or it can be achieved by doing the translations in the other order – that is, vector addition is commutative: v + w = w + v. Note that it makes sense to add two vectors in R2 , ...
... (v1 + w1 , v2 + w2 ) is the composition of doing the translation (v1 , w1 ) first and then doing the translation (v2 , w2 ) or it can be achieved by doing the translations in the other order – that is, vector addition is commutative: v + w = w + v. Note that it makes sense to add two vectors in R2 , ...
BANACH ALGEBRAS 1. Banach Algebras The aim of this notes is to
... Example 3.2. (1) Let z ∈ C. Then σ(z) = {z}. (2) Let A ∈ Mn (C). Then σ(A) = {λ ∈ C : λ is an eigen value of A} (3) Let f ∈ C(K) for some compact Hausdorff K. Then σ(f ) = range(f ) First we need to show that the spectrum of an element in a Banach algebra is a non empty compact set. Theorem 3.3. Let ...
... Example 3.2. (1) Let z ∈ C. Then σ(z) = {z}. (2) Let A ∈ Mn (C). Then σ(A) = {λ ∈ C : λ is an eigen value of A} (3) Let f ∈ C(K) for some compact Hausdorff K. Then σ(f ) = range(f ) First we need to show that the spectrum of an element in a Banach algebra is a non empty compact set. Theorem 3.3. Let ...
Factorization of unitary representations of adele groups
... Let A be a subalgebra of the algebra B(V ) of continuous linear operators on a Hilbert space V . Then the commutant A0 of A is defined to be A0 = {T ∈ B(V ) : T ◦ α = α ◦ T, for all α ∈ A} Schur’s Lemma asserts that, if (π, V ) is an irreducible unitary Hilbert space representation of a topological ...
... Let A be a subalgebra of the algebra B(V ) of continuous linear operators on a Hilbert space V . Then the commutant A0 of A is defined to be A0 = {T ∈ B(V ) : T ◦ α = α ◦ T, for all α ∈ A} Schur’s Lemma asserts that, if (π, V ) is an irreducible unitary Hilbert space representation of a topological ...
Study Guide Chapter 11
... normal vector N = (2,4,1). The equations of the planes can be written as N P = d. A typical point in N-P = 14 is (x, y, z) = (2,2,2) because then 22 4y z = 14. A typical point in N-P = 0 is (2,-1,O) because then 2x+4y+z=O. R o m N-P= 0 we see that N is perpendicular ( = normal ) to every vector P in ...
... normal vector N = (2,4,1). The equations of the planes can be written as N P = d. A typical point in N-P = 14 is (x, y, z) = (2,2,2) because then 22 4y z = 14. A typical point in N-P = 0 is (2,-1,O) because then 2x+4y+z=O. R o m N-P= 0 we see that N is perpendicular ( = normal ) to every vector P in ...
Classical and intuitionistic relation algebras
... under canonical extensions, and that a relation algebra is complete and atomic with all atoms as functional elements if and only if it is the complex algebra of a generalized Brandt groupoid. The results about canonical extensions were extended to distributive lattices with operators by Gehrke and J ...
... under canonical extensions, and that a relation algebra is complete and atomic with all atoms as functional elements if and only if it is the complex algebra of a generalized Brandt groupoid. The results about canonical extensions were extended to distributive lattices with operators by Gehrke and J ...
We assume all Lie algebras and vector spaces are finite
... The proof is a straightforward induction argument that uses corollary 1. 2. Abstract Jordan decomposition Let V be a finite dimensional vector space over the algebraically closed field k. The following is the abstract Jordan decomposition theorem for linear operators on V . Theorem 4. Let T be a lin ...
... The proof is a straightforward induction argument that uses corollary 1. 2. Abstract Jordan decomposition Let V be a finite dimensional vector space over the algebraically closed field k. The following is the abstract Jordan decomposition theorem for linear operators on V . Theorem 4. Let T be a lin ...
Math 311: Topics in Applied Math 1 3: Vector Spaces 3.2
... R3 , the vector space of all 3-element real column vectors over the real field R. (To save vertical space, we’ll denote column vectors as transposes of row vectors.) n ...
... R3 , the vector space of all 3-element real column vectors over the real field R. (To save vertical space, we’ll denote column vectors as transposes of row vectors.) n ...
vectors - MySolutionGuru
... Most of the quantities measured in science are classified as either scalars or vectors. Scalar : A scalar quantity is one which has only magnitude but no direction. Examples : Mass, time, speed, work, energy, volume, density etc. are scalars. Vectors : A vector quantity is one which has both magnitu ...
... Most of the quantities measured in science are classified as either scalars or vectors. Scalar : A scalar quantity is one which has only magnitude but no direction. Examples : Mass, time, speed, work, energy, volume, density etc. are scalars. Vectors : A vector quantity is one which has both magnitu ...
Holt Algebra 1 11-EXT
... There are inverse operations for other powers as well. For example 3 represents a cube root, and it is the inverse of cubing a number. To find 3 , look for three equal factors whose product is 8. Since 2 • 2 • 2 = 8. ...
... There are inverse operations for other powers as well. For example 3 represents a cube root, and it is the inverse of cubing a number. To find 3 , look for three equal factors whose product is 8. Since 2 • 2 • 2 = 8. ...
LINEAR COMBINATIONS AND SUBSPACES
... (2) If U is a nonempty finite subset of Rn then the span of U is a subspace of Rn and is called the subspace spanned or generated by U . The first property of a subspace listed above is often useful for showing that a certain set cannot be a subspace. For example the set {(1, 1)} is not a subspace o ...
... (2) If U is a nonempty finite subset of Rn then the span of U is a subspace of Rn and is called the subspace spanned or generated by U . The first property of a subspace listed above is often useful for showing that a certain set cannot be a subspace. For example the set {(1, 1)} is not a subspace o ...
x+y
... 2.3 Axiomatic Definition of Boolean Algebra • In 1937, Claude Shannon founded both digital computer and digital circuit design theory when, as a 21-year-old master's degree student at the Massachusetts Institute of Technology (MIT), he wrote his thesis demonstrating that electrical applications of ...
... 2.3 Axiomatic Definition of Boolean Algebra • In 1937, Claude Shannon founded both digital computer and digital circuit design theory when, as a 21-year-old master's degree student at the Massachusetts Institute of Technology (MIT), he wrote his thesis demonstrating that electrical applications of ...
Second midterm solutions
... We see that α ∈ V and α ∈ W , so α ∈ V ∩ W . Since u1 , . . . , uj is a basis for V ∩ W , α can be written in a unique way as a linear combination of u1 , . . . , uj . This same expression must be the unique way of writing α as a linear combination of the basis u1 , . . . , uj , w1 , . . . , w for ...
... We see that α ∈ V and α ∈ W , so α ∈ V ∩ W . Since u1 , . . . , uj is a basis for V ∩ W , α can be written in a unique way as a linear combination of u1 , . . . , uj . This same expression must be the unique way of writing α as a linear combination of the basis u1 , . . . , uj , w1 , . . . , w for ...
Oct. 3
... Let a' be a row vector and b a column vector, both being n-tuples, that is vectors having n elements: ...
... Let a' be a row vector and b a column vector, both being n-tuples, that is vectors having n elements: ...
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation of their boundaries—a choice of clockwise or counterclockwise.When regarded in this manner, the exterior product of two vectors is called a 2-blade. More generally, the exterior product of any number k of vectors can be defined and is sometimes called a k-blade. It lives in a space known as the kth exterior power. The magnitude of the resulting k-blade is the volume of the k-dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors.The exterior algebra, or Grassmann algebra after Hermann Grassmann, is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not just k-blades, but sums of k-blades; such a sum is called a k-vector. The k-blades, because they are simple products of vectors, are called the simple elements of the algebra. The rank of any k-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an associative algebra, which means that α ∧ (β ∧ γ) = (α ∧ β) ∧ γ for any elements α, β, γ. The k-vectors have degree k, meaning that they are sums of products of k vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a graded algebra.The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. Differential forms are mathematical objects that represent infinitesimal areas of infinitesimal parallelograms (and higher-dimensional bodies), and so can be integrated over surfaces and higher dimensional manifolds in a way that generalizes the line integrals from calculus. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms, and the pairing between the exterior algebra and its dual is given by the interior product.