Math 670 HW #2

... all of Λ(V ) by setting the inner product of homogeneous elements of different degrees equal to zero and by letting hw1 ∧ . . . ∧ wk , v1 ∧ . . . ∧ vk i = det (hwi , vj i)i,j and extending bilinearly. Since Λn (V ) is a one-dimensional real vector space, Λn (V ) − {0} has two components. An orientat ...

... all of Λ(V ) by setting the inner product of homogeneous elements of different degrees equal to zero and by letting hw1 ∧ . . . ∧ wk , v1 ∧ . . . ∧ vk i = det (hwi , vj i)i,j and extending bilinearly. Since Λn (V ) is a one-dimensional real vector space, Λn (V ) − {0} has two components. An orientat ...

Free associative algebras

... The point of these notes is to recall some linear algebra that we’ll be using in many forms in 18.745. You can think of the notes as a makeup for the canceled class February 10. Vector spaces can be thought of as a very nice place to study addition. The notion of direct sum of vector spaces provides ...

... The point of these notes is to recall some linear algebra that we’ll be using in many forms in 18.745. You can think of the notes as a makeup for the canceled class February 10. Vector spaces can be thought of as a very nice place to study addition. The notion of direct sum of vector spaces provides ...

LECTURE 21: SYMMETRIC PRODUCTS AND ALGEBRAS

... is a basis for S n (V ). Before we prove this, we note a key difference between this and the exterior product. For the exterior product, we have strict inequalities. For the symmetric product, we have non-strict inequalities. This makes a huge difference! Proof. We first show the set spans. Since {~ ...

... is a basis for S n (V ). Before we prove this, we note a key difference between this and the exterior product. For the exterior product, we have strict inequalities. For the symmetric product, we have non-strict inequalities. This makes a huge difference! Proof. We first show the set spans. Since {~ ...

2/4/15

... There is one more important definition we need before we can introduce the determinant. We will want to be able to look at the skew-symmetric tensor product: we want to look at the tensor product with one extra relation imposed: that v ⊗ v 0 = −v 0 ⊗ v. Note that this only makes sense if we’re looki ...

... There is one more important definition we need before we can introduce the determinant. We will want to be able to look at the skew-symmetric tensor product: we want to look at the tensor product with one extra relation imposed: that v ⊗ v 0 = −v 0 ⊗ v. Note that this only makes sense if we’re looki ...

Constructions in linear algebra For all that follows, let k be the base

... 2. Let W ⊂ V . Note that there is no canonical inclusion W ∗ ⊂ V ∗ . Let i : W → V be the inclusion; then there is a map i∗ : V ∗ → W ∗ . Show that i∗ is a quotient map, and show that its kernel is W ⊥ . ...

... 2. Let W ⊂ V . Note that there is no canonical inclusion W ∗ ⊂ V ∗ . Let i : W → V be the inclusion; then there is a map i∗ : V ∗ → W ∗ . Show that i∗ is a quotient map, and show that its kernel is W ⊥ . ...

Levi-Civita symbol

... follows similarly from equation 2. To establish equation 4, let us first observe that both sides vanish when . Indeed, if , then one can not choose m and n such that both permutation symbols on the left are nonzero. Then, with i = j fixed, there are only two ways to choose m and n from the remaining ...

... follows similarly from equation 2. To establish equation 4, let us first observe that both sides vanish when . Indeed, if , then one can not choose m and n such that both permutation symbols on the left are nonzero. Then, with i = j fixed, there are only two ways to choose m and n from the remaining ...

Solution

... pointwise. Thus f corresponds exactly to a vector in F ×∞ . Since F ×∞ is never isomorphic to F ⊕∞ we see that V ∗ is not always isomorphic to V . Problem 3: Suppose that F 0 is a field containing F and V is an F -vector space. If we consider F 0 to be an F -vector space, we can form the tensor prod ...

... pointwise. Thus f corresponds exactly to a vector in F ×∞ . Since F ×∞ is never isomorphic to F ⊕∞ we see that V ∗ is not always isomorphic to V . Problem 3: Suppose that F 0 is a field containing F and V is an F -vector space. If we consider F 0 to be an F -vector space, we can form the tensor prod ...

Chapter 2 - Cartesian Vectors and Tensors: Their Algebra Definition

... a × b = ε ijk ai b j e( k ) Velocity due to rigid body rotations We will show that the velocity field of a rigid body can be described by two vectors, a translation velocity, v(t), and an angular velocity, ω. A rigid body has the constraint that the distance between two points in the body does not c ...

... a × b = ε ijk ai b j e( k ) Velocity due to rigid body rotations We will show that the velocity field of a rigid body can be described by two vectors, a translation velocity, v(t), and an angular velocity, ω. A rigid body has the constraint that the distance between two points in the body does not c ...

Tensor Algebra: A Combinatorial Approach to the Projective Geometry of Figures

... All quadratic curves in the plane can be given in terms of this incidence relation by varying the coefficient vector cA(2) of the hypersurface. This concept can be used to define curves of any degree in P2 and likewise surfaces of any degree in P3 . The dualization operator for symmetrically embedd ...

... All quadratic curves in the plane can be given in terms of this incidence relation by varying the coefficient vector cA(2) of the hypersurface. This concept can be used to define curves of any degree in P2 and likewise surfaces of any degree in P3 . The dualization operator for symmetrically embedd ...

Introduction to tensor, tensor factorization and its applications

... processing, storage, tensor factorization is often needed. Three-way tensor: ...

... processing, storage, tensor factorization is often needed. Three-way tensor: ...

An Introduction to Nonlinear Solid Mechanics Marino Arroyo & Anna Pandolfi

... If Q and R are orthogonal, then also the product QR is orthogonal. It is said that the set of orthogonal mappings is a subgroup of GL(n) closed under multiplication, and it is called Orthogonal Group O(n). The subgroup of the orthogonal group that preserves the orientation, i.e. with determinant equ ...

... If Q and R are orthogonal, then also the product QR is orthogonal. It is said that the set of orthogonal mappings is a subgroup of GL(n) closed under multiplication, and it is called Orthogonal Group O(n). The subgroup of the orthogonal group that preserves the orientation, i.e. with determinant equ ...

Exploring Tensor Rank

... n1 × · · · × nd tensor is simply an n1 × · · · × nd -dimensional array of elements of a field F . We call d the order of the tensor, and (n1 , . . . , nd ) the dimensions of the tensor. The space of n1 × · · · × nd tensors is denoted F n1 ⊗ · · · ⊗ F nd , and is equipped with a natural addition wher ...

... n1 × · · · × nd tensor is simply an n1 × · · · × nd -dimensional array of elements of a field F . We call d the order of the tensor, and (n1 , . . . , nd ) the dimensions of the tensor. The space of n1 × · · · × nd tensors is denoted F n1 ⊗ · · · ⊗ F nd , and is equipped with a natural addition wher ...

THE TENSOR PRODUCT OF FUNCTION SEMIMODULES The

... such that µ : SA ⊗ SB → SA×B is not injective (Theorem 3.10). Here, ⊗ denotes the (algebraic) tensor product of S-semimodules satisfying the standard universal property expected of such a product. This tensor product was constructed by Y. Katsov in [Kat97]. The infinite cardinality of A, B has to be ...

... such that µ : SA ⊗ SB → SA×B is not injective (Theorem 3.10). Here, ⊗ denotes the (algebraic) tensor product of S-semimodules satisfying the standard universal property expected of such a product. This tensor product was constructed by Y. Katsov in [Kat97]. The infinite cardinality of A, B has to be ...

TENSOR PRODUCTS OF LOCALLY CONVEX ALGEBRAS 124

... Banach algebras with respect to a cross-norm, these authors have given sufficient conditions under which its space of nonzero multiplicative functionals with relative weak* topology can be characterized as the topological direct product of the corresponding spaces associated with the two factors. Gi ...

... Banach algebras with respect to a cross-norm, these authors have given sufficient conditions under which its space of nonzero multiplicative functionals with relative weak* topology can be characterized as the topological direct product of the corresponding spaces associated with the two factors. Gi ...

Chapter 3 Cartesian Tensors

... consider the vector relation y = (a . b)x. We have a . b = ai bi , but we cannot write yi = ai bi xi as this would be ambiguous. How can we correct this? Note that a . b = ai b i = aj b j – the suffix we use for the summation is immaterial. (Compare with the use of dummy R∞ R∞ variables in integrat ...

... consider the vector relation y = (a . b)x. We have a . b = ai bi , but we cannot write yi = ai bi xi as this would be ambiguous. How can we correct this? Note that a . b = ai b i = aj b j – the suffix we use for the summation is immaterial. (Compare with the use of dummy R∞ R∞ variables in integrat ...

Harmonic analysis of dihedral groups

... The rotations are the symmetries preserving the (cyclic) ordering of vertices. Thus, a rotation g is determined by the image gv, so the subgroup N of rotations has n elements. A reflection is an order-2 symmetry reversing the ordering of vertices. Imbedding the n-gon in R2 , there are n axes through ...

... The rotations are the symmetries preserving the (cyclic) ordering of vertices. Thus, a rotation g is determined by the image gv, so the subgroup N of rotations has n elements. A reflection is an order-2 symmetry reversing the ordering of vertices. Imbedding the n-gon in R2 , there are n axes through ...

Tensor products in the category of topological vector spaces are not

... For example, the tensor products in the class of real Hausdorff locally convex spaces are the projective tensor products, going back to Grothendieck’s memoir [8]. In this case, an explicit description of the locally convex topology (by means of suitable cross-seminorms) is available, and it is well- ...

... For example, the tensor products in the class of real Hausdorff locally convex spaces are the projective tensor products, going back to Grothendieck’s memoir [8]. In this case, an explicit description of the locally convex topology (by means of suitable cross-seminorms) is available, and it is well- ...

Part C4: Tensor product

... (3) list of basic properties (4) distributive property (5) right exactness (6) localization is flat (7) extension of scalars (8) applications 4.1. definition. First I gave the categorical definition and then I gave an explicit construction. 4.1.1. universal condition. Tensor product is usually defin ...

... (3) list of basic properties (4) distributive property (5) right exactness (6) localization is flat (7) extension of scalars (8) applications 4.1. definition. First I gave the categorical definition and then I gave an explicit construction. 4.1.1. universal condition. Tensor product is usually defin ...

Explicit tensors - Computational Complexity

... so we need to be able to encode the field elements by {0, 1}-strings. For instance, elements from finite fields can be represented in binary and rational numbers by tuples of integers represented in binary. The actual encoding does not matter as long as it is “reasonably nice”, that is, all operatio ...

... so we need to be able to encode the field elements by {0, 1}-strings. For instance, elements from finite fields can be represented in binary and rational numbers by tuples of integers represented in binary. The actual encoding does not matter as long as it is “reasonably nice”, that is, all operatio ...

MULTILINEAR ALGEBRA: THE EXTERIOR PRODUCT This writeup

... The argument for the remaining elementary divisors is more of the same, going up to F ∧m = Ae1 ∧ · · · ∧ em ⊕ · · · and S ∧m = a1 · · · am e1 ∧ · · · ∧ em . Note that m itself is described intrinsically as the highest exponent of a nonzero exterior power of S. Now we can see that the ingredients of ...

... The argument for the remaining elementary divisors is more of the same, going up to F ∧m = Ae1 ∧ · · · ∧ em ⊕ · · · and S ∧m = a1 · · · am e1 ∧ · · · ∧ em . Note that m itself is described intrinsically as the highest exponent of a nonzero exterior power of S. Now we can see that the ingredients of ...

TENSOR PRODUCTS II 1. Introduction Continuing our study of

... makes no sense in I ⊗2 since 1 is not an element of I.) To show two tensors are not equal, the best approach is to construct a linear map from the tensor product space that has different values at the two tensors. The function I × I → A given by (f, g) 7→ fX (0, 0)gY (0, 0), where fX and gY are part ...

... makes no sense in I ⊗2 since 1 is not an element of I.) To show two tensors are not equal, the best approach is to construct a linear map from the tensor product space that has different values at the two tensors. The function I × I → A given by (f, g) 7→ fX (0, 0)gY (0, 0), where fX and gY are part ...

Chapter 22 Tensor Algebras, Symmetric Algebras and Exterior

... isomorphism between T1 and T2 . Now that we have shown that tensor products are unique up to isomorphism, we give a construction that produces one. Theorem 22.5 Given n ≥ 2 vector spaces E1 , . . . , En , a tensor product (E1 ⊗ · · · ⊗ En , ϕ) for E1 , . . . , En can be constructed. Furthermore, den ...

... isomorphism between T1 and T2 . Now that we have shown that tensor products are unique up to isomorphism, we give a construction that produces one. Theorem 22.5 Given n ≥ 2 vector spaces E1 , . . . , En , a tensor product (E1 ⊗ · · · ⊗ En , ϕ) for E1 , . . . , En can be constructed. Furthermore, den ...

Square Deal: Lower Bounds and Improved Relaxations for Tensor

... ranktc (X 0 )} is empty.2 The recovery performance of (2.1) depends heavily on the properties of G. Suppose (2.1) fails to recover X 0 ∈ Tr . Then there exists another X 0 ∈ Tr such that G[X 0 ] = G[X 0 ]. To guarantee that (2.1) recovers any X 0 ∈ Tr , a necessary and sufficient condition is that G ...

... ranktc (X 0 )} is empty.2 The recovery performance of (2.1) depends heavily on the properties of G. Suppose (2.1) fails to recover X 0 ∈ Tr . Then there exists another X 0 ∈ Tr such that G[X 0 ] = G[X 0 ]. To guarantee that (2.1) recovers any X 0 ∈ Tr , a necessary and sufficient condition is that G ...

Tensors and hypermatrices

... useful device. First, it gives us a concrete way to think about tensors, one that allows a 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 ...

... useful device. First, it gives us a concrete way to think about tensors, one that allows a 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 ...