eiilm university, sikkim
... be added together and multiplied ("scaled") by numbers, calledscalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar mu ...
... be added together and multiplied ("scaled") by numbers, calledscalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar mu ...
CM222A LINEAR ALGEBRA Solutions 1 1. Determine whether the
... spanning, w = 1 αi vi for some scalars αi and w − 1 αi vi = 0 shows that T is linearly dependent. Conversely, if S satisfies (i) and (ii) it is linearly independent. To show that it is spanning, let v be any vector.PIf v ∈ S then clearly v ∈ span S. If v ∈ / S then S ∪ {v} is linearly dependent so f ...
... spanning, w = 1 αi vi for some scalars αi and w − 1 αi vi = 0 shows that T is linearly dependent. Conversely, if S satisfies (i) and (ii) it is linearly independent. To show that it is spanning, let v be any vector.PIf v ∈ S then clearly v ∈ span S. If v ∈ / S then S ∪ {v} is linearly dependent so f ...
A QUANTUM ANALOGUE OF KOSTANT`S THEOREM FOR THE
... is the following quantum analogue of Kostant’s classical theorem: Theorem 3.1.1. For q ∈ C× not a root of unity or q = 1, A is a free graded left I-module. Remark 3.1.2. The condition on q in the hypothesis of the theorem is needed only for the application of Theorem 2.3.1. In [AZ], Theorem 2.3.1 is ...
... is the following quantum analogue of Kostant’s classical theorem: Theorem 3.1.1. For q ∈ C× not a root of unity or q = 1, A is a free graded left I-module. Remark 3.1.2. The condition on q in the hypothesis of the theorem is needed only for the application of Theorem 2.3.1. In [AZ], Theorem 2.3.1 is ...
The graph planar algebra embedding
... embedding map. In the case that Γ is simply-laced (that is HomC (Q, P ⊗ X) is always 1-dimensional) choosing a basis is just a choice of normalization: note that these choices nevertheless affect the embedding map below. The principal graph splits into components according to the region label at the ...
... embedding map. In the case that Γ is simply-laced (that is HomC (Q, P ⊗ X) is always 1-dimensional) choosing a basis is just a choice of normalization: note that these choices nevertheless affect the embedding map below. The principal graph splits into components according to the region label at the ...
Latest Revision 09/21/06
... definition of matrix seems simple enough: a collection or an array of elements. At the secondary school level, we pay attention to matrices whose elements are real numbers, or maybe even rational numbers to be more specific. While matrices are composed of real numbers, not all properties that work f ...
... definition of matrix seems simple enough: a collection or an array of elements. At the secondary school level, we pay attention to matrices whose elements are real numbers, or maybe even rational numbers to be more specific. While matrices are composed of real numbers, not all properties that work f ...
L.L. STACHÓ- B. ZALAR, Bicircular projections in some matrix and
... norm and can change if an equivalent norm is taken instead. We shall work with some matrix spaces so we note that we always consider any space of matrices to be equipped with the spectral norm so we view Mn (C) as a special case of the algebra B(H ) of all bounded operators on a Hilbert space. It is ...
... norm and can change if an equivalent norm is taken instead. We shall work with some matrix spaces so we note that we always consider any space of matrices to be equipped with the spectral norm so we view Mn (C) as a special case of the algebra B(H ) of all bounded operators on a Hilbert space. It is ...
Lecture 3: Vector subspaces, sums, and direct sums (1)
... functions {0, 1, . . . , n − 1} → C? (a) All functions such that f (3) = 2f (1) (b) All functions such that f (5) = f (6) = 0 (c) All functions such that f (2) − f (0) = 1 (d) All functions such that f (2)f (5) = 0 Answer: (a): yes: if f and g satisfy these properties, so do f + g and af for all a ∈ ...
... functions {0, 1, . . . , n − 1} → C? (a) All functions such that f (3) = 2f (1) (b) All functions such that f (5) = f (6) = 0 (c) All functions such that f (2) − f (0) = 1 (d) All functions such that f (2)f (5) = 0 Answer: (a): yes: if f and g satisfy these properties, so do f + g and af for all a ∈ ...
The product topology.
... given, for each Q i ∈ I, a continuous mapping fi : A → Xi . Then there is a unique continuous Q map f : A → i Xi such that πi ◦ f = fi for all i ∈ I. Conversely, if f : A → i Xi is continuous, then each of the maps fi = πi ◦ f is continuous. Q Proof. The requirement that πi ◦ f = fi forces f (a)(i) ...
... given, for each Q i ∈ I, a continuous mapping fi : A → Xi . Then there is a unique continuous Q map f : A → i Xi such that πi ◦ f = fi for all i ∈ I. Conversely, if f : A → i Xi is continuous, then each of the maps fi = πi ◦ f is continuous. Q Proof. The requirement that πi ◦ f = fi forces f (a)(i) ...
HERE
... definition of matrix seems simple enough: a collection or an array of elements. At the secondary school level, we pay attention to matrices whose elements are real numbers, or maybe even rational numbers to be more specific. While matrices are composed of real numbers, not all properties that work f ...
... definition of matrix seems simple enough: a collection or an array of elements. At the secondary school level, we pay attention to matrices whose elements are real numbers, or maybe even rational numbers to be more specific. While matrices are composed of real numbers, not all properties that work f ...
Chapter 6 General Linear Transformations
... We studied linear transformations in Section 4.2 and used the following definition to determine whether a transformation from Rn to Rm is linear. Definition 6.1.1. A transformation T : Rn → Rm is linear if the following two properties hold for all vectors ~u and ~v in Rn and every scalar c: (a) T (~ ...
... We studied linear transformations in Section 4.2 and used the following definition to determine whether a transformation from Rn to Rm is linear. Definition 6.1.1. A transformation T : Rn → Rm is linear if the following two properties hold for all vectors ~u and ~v in Rn and every scalar c: (a) T (~ ...
Elementary Algebra
... All courses based on the academic standards for elementary algebra must include instruction using the mathematics process standards, allowing students to engage in problem solving, decision making, critical thinking, and applied learning. Educators must determine the extent to which such courses or ...
... All courses based on the academic standards for elementary algebra must include instruction using the mathematics process standards, allowing students to engage in problem solving, decision making, critical thinking, and applied learning. Educators must determine the extent to which such courses or ...
Note
... The null space of A N(A) is the same as the null space of U. If the system A.x = 0 is reduced to U.x = 0 then none of the solutions (which constitute the null space) is changed. It has dimension n – r. If homogeneous solutions to U.x = 0 are found, they constitute a basis for N(A). The column space ...
... The null space of A N(A) is the same as the null space of U. If the system A.x = 0 is reduced to U.x = 0 then none of the solutions (which constitute the null space) is changed. It has dimension n – r. If homogeneous solutions to U.x = 0 are found, they constitute a basis for N(A). The column space ...
General Vector Spaces I
... 1. Recognize from the standard examples of vector spaces, that a vector space is closed under vector addition and scalar multiplication. 2. Determine if a subset W of a vector space V is a subspace of V. 3. Find the linear combination of a finite set of vectors. 4. Find W = span(S), a subspace of V, ...
... 1. Recognize from the standard examples of vector spaces, that a vector space is closed under vector addition and scalar multiplication. 2. Determine if a subset W of a vector space V is a subspace of V. 3. Find the linear combination of a finite set of vectors. 4. Find W = span(S), a subspace of V, ...
Frobenius algebras and monoidal categories
... An algebra A over a field k is called Frobenius when it is finite dimensional and equipped with a linear function e : A æ æÆ k such that: e(ab) = 0 for all a Œ A implies b = 0. ...
... An algebra A over a field k is called Frobenius when it is finite dimensional and equipped with a linear function e : A æ æÆ k such that: e(ab) = 0 for all a Œ A implies b = 0. ...
Rohit Yalamati - The Product Topology
... Euclidean Space: Euclidean nspace, sometimes called Cartesian space or simply n space, is the space of all ntuples of real numbers, (x1, x2, ..., xn) ...
... Euclidean Space: Euclidean nspace, sometimes called Cartesian space or simply n space, is the space of all ntuples of real numbers, (x1, x2, ..., xn) ...
ON NONASSOCIATIVE DIVISION ALGEBRAS^)
... License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ...
... License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ...
p. 1 Math 490 Notes 11 Initial Topologies: Subspaces and Products
... Theorem N11.1 In the notation of the previous definition, τ is the coarsest topology on X relative to which fi : (X, τ ) → (Xi , τi ) is continuous for all i ∈ I. Proof : To show that fi is continuous for every i, note that Ui ∈ τi ⇒ f −1 (Ui ) ∈ τ by definition. To show that τ is the coarsest such ...
... Theorem N11.1 In the notation of the previous definition, τ is the coarsest topology on X relative to which fi : (X, τ ) → (Xi , τi ) is continuous for all i ∈ I. Proof : To show that fi is continuous for every i, note that Ui ∈ τi ⇒ f −1 (Ui ) ∈ τ by definition. To show that τ is the coarsest such ...
A v
... {v1, v2, …, vn} are linearly independent {v1, v2, …, vn} span the whole vector space V: V = {1v1 + 2v2 + … + nvn | i is scalar} Any vector in V is a unique linear combination of the basis. The number of basis vectors is called the dimension of V. ...
... {v1, v2, …, vn} are linearly independent {v1, v2, …, vn} span the whole vector space V: V = {1v1 + 2v2 + … + nvn | i is scalar} Any vector in V is a unique linear combination of the basis. The number of basis vectors is called the dimension of V. ...
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.