Linear Algebra and Matrices
... Vectorial space: space defined by different vectors (for example for dimensions…). The vectorial space defined by some vectors is a space that contains them and all the vectors that can be obtained by multiplying these vectors by a real number then adding them (linear combination). A matrix A (mn) ...
... Vectorial space: space defined by different vectors (for example for dimensions…). The vectorial space defined by some vectors is a space that contains them and all the vectors that can be obtained by multiplying these vectors by a real number then adding them (linear combination). A matrix A (mn) ...
MATH 307 Subspaces
... What if the line did not go through the origin? Consider U = {( x , y ) : y = m x + b } € c ≠ 1, c y = c (m x + b) = m(c x) + c b ≠ m x + b ; for some b ≠ 0. Then for ( x , y ) ∈ U and hence, U is not closed under scalar multiplication. It also can be shown that U is not closed under addition; but o ...
... What if the line did not go through the origin? Consider U = {( x , y ) : y = m x + b } € c ≠ 1, c y = c (m x + b) = m(c x) + c b ≠ m x + b ; for some b ≠ 0. Then for ( x , y ) ∈ U and hence, U is not closed under scalar multiplication. It also can be shown that U is not closed under addition; but o ...
pdf file on-line
... U D1 U = D2 If there exist grading operators γ1 , γ2 then we also demand that U γ1 U ∗ = γ2 . If there exist real structures J1 , J2 then we also demand that U J1 U ∗ = J2 . However, following the spirit of our extended notion of morphisms between algebras, it is more natural to have equivalences as ...
... U D1 U = D2 If there exist grading operators γ1 , γ2 then we also demand that U γ1 U ∗ = γ2 . If there exist real structures J1 , J2 then we also demand that U J1 U ∗ = J2 . However, following the spirit of our extended notion of morphisms between algebras, it is more natural to have equivalences as ...
Invertible and nilpotent elements in the group algebra of a
... Our approach uses a little bit of algebraic geometry and might possibly also be of interest to solve other problems related to group algebras. It is inspired by a recent result of Ottmar Loos in [4], where he determines the invertible elements in a Laurent polynomial ring k[t±1 ]. In Th. 3 we descri ...
... Our approach uses a little bit of algebraic geometry and might possibly also be of interest to solve other problems related to group algebras. It is inspired by a recent result of Ottmar Loos in [4], where he determines the invertible elements in a Laurent polynomial ring k[t±1 ]. In Th. 3 we descri ...
Linearly independence Definition: Consider a set of n
... Definition: The minimum number of linearly independent vector to span a space is the dimension of the vector space. In the above example, the vector space V 3 (R) has a dimension of 3, because that is the minimum of linearly independent vectors that is required to span it. Definition: Consider a ve ...
... Definition: The minimum number of linearly independent vector to span a space is the dimension of the vector space. In the above example, the vector space V 3 (R) has a dimension of 3, because that is the minimum of linearly independent vectors that is required to span it. Definition: Consider a ve ...
Vector Space
... vector ez points in the z direction. There are several common notations for these vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. In addition, these vectors are sometimes written with a hat to emphasize their status as unit vectors. These vectors are a basis in the sense tha ...
... vector ez points in the z direction. There are several common notations for these vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. In addition, these vectors are sometimes written with a hat to emphasize their status as unit vectors. These vectors are a basis in the sense tha ...
Chapter 4: Lie Algebras
... Lie algebras are constructed by linearizing Lie groups. A Lie group can be linearized in the neighborhood of any of its points, or group operations. Linearization amounts to Taylor series expansion about the coordinates that define the group operation. What is being Taylor expanded is the group comp ...
... Lie algebras are constructed by linearizing Lie groups. A Lie group can be linearized in the neighborhood of any of its points, or group operations. Linearization amounts to Taylor series expansion about the coordinates that define the group operation. What is being Taylor expanded is the group comp ...
Spin structures
... Theorem 10. Let E be an oriented vector bundle over a manifold X. Then there exists a spin structure on E if and only if the second Stiefel-Whitney class of E is zero. Furthermore, if w2(E) = 0, then the distinct spin structures on E are in 1-to-1 correspondence with the elements of H 1(X, Z2). Defi ...
... Theorem 10. Let E be an oriented vector bundle over a manifold X. Then there exists a spin structure on E if and only if the second Stiefel-Whitney class of E is zero. Furthermore, if w2(E) = 0, then the distinct spin structures on E are in 1-to-1 correspondence with the elements of H 1(X, Z2). Defi ...
n-ARY LIE AND ASSOCIATIVE ALGEBRAS Peter W. Michor
... Apparently he looked for a simple model which explains the unseparability of quarks. Much later, in the early 90’s, it was noticed by M. Flato, C. Fronsdal, and others, that the n-bracket (2) satisfies (1). On this basis L. Takhtajan [17] developed sytematically the foundations of of the theory of n ...
... Apparently he looked for a simple model which explains the unseparability of quarks. Much later, in the early 90’s, it was noticed by M. Flato, C. Fronsdal, and others, that the n-bracket (2) satisfies (1). On this basis L. Takhtajan [17] developed sytematically the foundations of of the theory of n ...
Classical groups and their real forms
... Let us consider differentiable curves σ : (−ε, ε) −→ GL(V ) such that σ(0) = I and σ(t) ∈ G, e.g. σ(t) satisfies the conditions which defined G as a subgroup of GL(V ). Now we have shown that σ 0 (0) ∈ Lie(G) and that every element in the lie algebra is obtained like this. We can now consider Isom(V, ...
... Let us consider differentiable curves σ : (−ε, ε) −→ GL(V ) such that σ(0) = I and σ(t) ∈ G, e.g. σ(t) satisfies the conditions which defined G as a subgroup of GL(V ). Now we have shown that σ 0 (0) ∈ Lie(G) and that every element in the lie algebra is obtained like this. We can now consider Isom(V, ...
Contents Definition of a Subspace of a Vector Space
... Matrices A and B are symmetric if A = AT and B = BT. Then (A + B)T = AT + BT = A + B so A + B is also symmetric. In addition, (rA)T = rAT = rA and so rA is symmetric. Hence the set of symmetric matrices is closed under addition and scalar multiplication. ...
... Matrices A and B are symmetric if A = AT and B = BT. Then (A + B)T = AT + BT = A + B so A + B is also symmetric. In addition, (rA)T = rAT = rA and so rA is symmetric. Hence the set of symmetric matrices is closed under addition and scalar multiplication. ...
Tensors and hypermatrices
... parallel to the usual matrix theory. Second, a hypermatrix is what we often get in practice: As soon as measurements are performed in some units, bases are chosen implicitly, and the values of the measurements are then recorded in the form of a hypermatrix. (There are of course good reasons not to j ...
... parallel to the usual matrix theory. Second, a hypermatrix is what we often get in practice: As soon as measurements are performed in some units, bases are chosen implicitly, and the values of the measurements are then recorded in the form of a hypermatrix. (There are of course good reasons not to j ...
An Introduction to Nonlinear Solid Mechanics Marino Arroyo & Anna Pandolfi
... Symmetric Tensors, Spectral Decomposition ...
... Symmetric Tensors, Spectral Decomposition ...
Vector Spaces - Beck-Shop
... Example 2.10 Yml (θ, φ) We saw in the previous section that the Yml are elements of H̃l , which can be obtained from Hl (R3 ) by restricting to the unit sphere. What is more is that the set {Yml }−l≤m≤l is actually a basis for H̃l . In the case l = 1 this is clear: we have H1 (R3 ) = P1 (R3 ) and cl ...
... Example 2.10 Yml (θ, φ) We saw in the previous section that the Yml are elements of H̃l , which can be obtained from Hl (R3 ) by restricting to the unit sphere. What is more is that the set {Yml }−l≤m≤l is actually a basis for H̃l . In the case l = 1 this is clear: we have H1 (R3 ) = P1 (R3 ) and cl ...
PPT - Jung Y. Huang
... You don’t necessarily have to be able to multiply two vectors by each other or even to be able to define the length of a vector, though those are very useful operations. The common example of directed line segments (arrows) in 2D or 3D fits this idea, because you can add such arrows by the parallelo ...
... You don’t necessarily have to be able to multiply two vectors by each other or even to be able to define the length of a vector, though those are very useful operations. The common example of directed line segments (arrows) in 2D or 3D fits this idea, because you can add such arrows by the parallelo ...
Topology Change for Fuzzy Physics: Fuzzy Spaces as Hopf Algebras
... Fuzzy spaces provide finite-dimensional approximations to certain symplectic manifolds M such as S 2 ≃ CP 1 , S 2 × S 2 and CP 2 . They are typically full matrix algebras M at(N + 1) of dimension (N + 1) × (N + 1). The fuzzy sphere SF2 (J) for angular momentum J = N2 for example is M at(N + 1). As N ...
... Fuzzy spaces provide finite-dimensional approximations to certain symplectic manifolds M such as S 2 ≃ CP 1 , S 2 × S 2 and CP 2 . They are typically full matrix algebras M at(N + 1) of dimension (N + 1) × (N + 1). The fuzzy sphere SF2 (J) for angular momentum J = N2 for example is M at(N + 1). As N ...
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.