CSCE 590E Spring 2007
... two vectors that produces a scalar The dot product between two n-dimensional vectors V and W is ...
... two vectors that produces a scalar The dot product between two n-dimensional vectors V and W is ...
CBrayMath216-2-4-f.mp4 SPEAKER: We`re quickly approaching
... differential equations. [INAUDIBLE] So we will use those subsequent linear algebra for this other tremendous application of solving a large category of differential equations. So now that we are near the end of our first segment of the course, you might say, I want to do a quick summary of related k ...
... differential equations. [INAUDIBLE] So we will use those subsequent linear algebra for this other tremendous application of solving a large category of differential equations. So now that we are near the end of our first segment of the course, you might say, I want to do a quick summary of related k ...
Lecture 1 - Lie Groups and the Maurer-Cartan equation
... algebra of left-invariant vector fields on the manifold G. Since this is a Lie subalgebra of the Lie algebra of all differentiable vector fields under the bracket, the Jacobi identity and antisymmetry hold, so we have a lie algebra g canonically associated with the group G, with dim g = dim G. We h ...
... algebra of left-invariant vector fields on the manifold G. Since this is a Lie subalgebra of the Lie algebra of all differentiable vector fields under the bracket, the Jacobi identity and antisymmetry hold, so we have a lie algebra g canonically associated with the group G, with dim g = dim G. We h ...
Fall 1993 MA Comprehensive Exam in Algebra
... 5. Let V be a finite dimensional vector space over C and T : V → V be a linear transformation satisfying the equation T 4 = I. (a) Prove that T can be represented by a diagonal matrix. (b) Give an example to show that if V is a finite dimensional vector space over R, and T is as above, then T need ...
... 5. Let V be a finite dimensional vector space over C and T : V → V be a linear transformation satisfying the equation T 4 = I. (a) Prove that T can be represented by a diagonal matrix. (b) Give an example to show that if V is a finite dimensional vector space over R, and T is as above, then T need ...
X. A brief review of linear vector spaces
... concern is the matrix G. We examine the structure of G to gain insight into the problem we have posed. It is wise to remember that our investigation into G has only to do with our model, and has nothing to do with the data. We will use the language of linear algebra to discuss how G is a transformat ...
... concern is the matrix G. We examine the structure of G to gain insight into the problem we have posed. It is wise to remember that our investigation into G has only to do with our model, and has nothing to do with the data. We will use the language of linear algebra to discuss how G is a transformat ...
Lec 31: Inner products. An inner product on a vector space V
... (1) fails. Then A is not a matrix of inner product. Hence the formula (u, v) = uT Av (with symmetric A) defines an inner product in Rn , if and only if (u, u) = uT Au is positive for all nonzero u (so, (1) is satisfied). Such symmetric matrices A are called positive definite. Thus, positive definite ...
... (1) fails. Then A is not a matrix of inner product. Hence the formula (u, v) = uT Av (with symmetric A) defines an inner product in Rn , if and only if (u, u) = uT Au is positive for all nonzero u (so, (1) is satisfied). Such symmetric matrices A are called positive definite. Thus, positive definite ...
Universal enveloping algebra
... Example 17.3.2. One important example is the case when A = EndF (V ) is the algebra of F -linear endomorphisms of a vector space V . Then L(A) = gl(V ) and ϕ : L → L(A) = gl(V ) is a representation of L making V into an L-module. The algebra homomorphism ψ : U → A = EndF (V ) makes V into a module o ...
... Example 17.3.2. One important example is the case when A = EndF (V ) is the algebra of F -linear endomorphisms of a vector space V . Then L(A) = gl(V ) and ϕ : L → L(A) = gl(V ) is a representation of L making V into an L-module. The algebra homomorphism ψ : U → A = EndF (V ) makes V into a module o ...
APPM 2360 17 October, 2013 Worksheet #7 1. Consider the space
... ii. ∀λ ∈ R we get λu = λf (x), since λf (i) = λ0 = 0 then λu ∈ W Since set W satisfies both conditions of subspace definition, W is subspace of the space of all polynomials with real coefficients. 2. (a) Any subset of a vector space is also a vector space. (b) A linearly independent set of vectors i ...
... ii. ∀λ ∈ R we get λu = λf (x), since λf (i) = λ0 = 0 then λu ∈ W Since set W satisfies both conditions of subspace definition, W is subspace of the space of all polynomials with real coefficients. 2. (a) Any subset of a vector space is also a vector space. (b) A linearly independent set of vectors i ...
1. New Algebraic Tools for Classical Geometry
... Classical geometry has emerged from efforts to codify perception of space and motion. With roots in ancient times, the great flowering of classical geometry was in the 19th century, when Euclidean, non-Euclidean and projective geometries were given precise mathematical formulations and the rich prop ...
... Classical geometry has emerged from efforts to codify perception of space and motion. With roots in ancient times, the great flowering of classical geometry was in the 19th century, when Euclidean, non-Euclidean and projective geometries were given precise mathematical formulations and the rich prop ...
(pdf)
... β : W → X are K-linear maps, then (β ◦ α) = α∗ ◦ β ∗ . Proof. If α is a K-linear map, then α∗ (f1 + f2 ) = (f1 + f2 ) ◦ α = f1 ◦ α + f2 ◦ α = α∗ (f1 ) + α∗ (f2 ) and α∗ (af ) = af ◦ α = aα∗ (f ) for all f , f1 , f2 ∈ W ∗ , a ∈ K, so α∗ is a K-linear map. For any map f : X → K, ...
... β : W → X are K-linear maps, then (β ◦ α) = α∗ ◦ β ∗ . Proof. If α is a K-linear map, then α∗ (f1 + f2 ) = (f1 + f2 ) ◦ α = f1 ◦ α + f2 ◦ α = α∗ (f1 ) + α∗ (f2 ) and α∗ (af ) = af ◦ α = aα∗ (f ) for all f , f1 , f2 ∈ W ∗ , a ∈ K, so α∗ is a K-linear map. For any map f : X → K, ...
Properties of lengths and dis- tances Orthogonal complements
... Notice that both spaces V and V ∗ , both being of dimension n, are of course isomorphic, as are all vector spaces of the same dimension. Such an isomorphism can be constructed by a random choice of bases in V and V ∗ and identifying both spaces with Rn . It is does not require any Euclidean structur ...
... Notice that both spaces V and V ∗ , both being of dimension n, are of course isomorphic, as are all vector spaces of the same dimension. Such an isomorphism can be constructed by a random choice of bases in V and V ∗ and identifying both spaces with Rn . It is does not require any Euclidean structur ...
SG 10 Basic Algebra
... In basic algebra, letters represent numbers. It is important to collect same letters together when possible. For example: ...
... In basic algebra, letters represent numbers. It is important to collect same letters together when possible. For example: ...
33-759 Introduction to Mathematical Physics Fall Semester, 2005 Assignment No. 8.
... 1. Each of the following statements is almost, but not quite, correct. In each case find the (or at leaat a) correct statement by making a (small) change in the original statement, or perhaps adding a qualification. While doing this, or in addition, indicate what was wrong with the original statemen ...
... 1. Each of the following statements is almost, but not quite, correct. In each case find the (or at leaat a) correct statement by making a (small) change in the original statement, or perhaps adding a qualification. While doing this, or in addition, indicate what was wrong with the original statemen ...
HILBERT SPACE GEOMETRY
... vector and that it is closed under vector addition and scalar multiplication: M=∅ ⇒ span(M)={0} ⇒ 0∈span(M) M≠∅ ⇒ ∃x∈M: 0⋅x=0∈span(M) x,y∈span(M) ⇒ x+y=1⋅x+1⋅y∈span(M) x∈span(M), λ∈ ⇒ λ⋅x∈span(M) The other properties of a vector space are satisfied for all elements of V and therefore also for all el ...
... vector and that it is closed under vector addition and scalar multiplication: M=∅ ⇒ span(M)={0} ⇒ 0∈span(M) M≠∅ ⇒ ∃x∈M: 0⋅x=0∈span(M) x,y∈span(M) ⇒ x+y=1⋅x+1⋅y∈span(M) x∈span(M), λ∈ ⇒ λ⋅x∈span(M) The other properties of a vector space are satisfied for all elements of V and therefore also for all el ...
Matrix product. Let A be an m × n matrix. If x ∈ IR is a
... defines the matrix product function f (x) = Ax from IRn to IRm . Similarly, if B is an n × p matrix and y ∈ IRp is a (column) vector, then p X By = x : bkj yj = xk , 1 ≤ k ≤ n j=1 ...
... defines the matrix product function f (x) = Ax from IRn to IRm . Similarly, if B is an n × p matrix and y ∈ IRp is a (column) vector, then p X By = x : bkj yj = xk , 1 ≤ k ≤ n j=1 ...
intuition
... • MRA (multiresolution analysis) – Construct a hierarchy of approximations to functions in various subspaces of a linear vector space ...
... • MRA (multiresolution analysis) – Construct a hierarchy of approximations to functions in various subspaces of a linear vector space ...
Updated 1/26/17 Amanda Havens 530-514-9373 Jr
... Dept. of Math Statistics – Private Tutor – Spring 2017 The tutors on this list are working as independent contractors. They are not hired by the Dept. of Math Statistics at CSU, Chico. Please contact them directly. ...
... Dept. of Math Statistics – Private Tutor – Spring 2017 The tutors on this list are working as independent contractors. They are not hired by the Dept. of Math Statistics at CSU, Chico. Please contact them directly. ...
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.