X - JP McCarthy: Math Page
... variables is replaced by a more general algebra. One particular question of classical probability is “How many shuffles does it take to mix up a deck of cards?” and can be answered in the language of random walks on groups. There is the notion of a random walk on a quantum group, and I am trying to ...
... variables is replaced by a more general algebra. One particular question of classical probability is “How many shuffles does it take to mix up a deck of cards?” and can be answered in the language of random walks on groups. There is the notion of a random walk on a quantum group, and I am trying to ...
PDF
... In the foregoing discussion, an algebra shall mean a non-associative algebra. Let A be a normed ∗-algebra, an algebra admitting an involution ∗, over a commutative ring R with 1 6= 0. The Cayley-Dickson construction is a way of enlarging A to a new algebra, KD(A), extending the ∗ as well as the norm ...
... In the foregoing discussion, an algebra shall mean a non-associative algebra. Let A be a normed ∗-algebra, an algebra admitting an involution ∗, over a commutative ring R with 1 6= 0. The Cayley-Dickson construction is a way of enlarging A to a new algebra, KD(A), extending the ∗ as well as the norm ...
Sol - Math TAMU
... Sol. Let V denote the set of vectors of the above form. By rewriting [a, b, 0, c, a − 2b + c] = a[1, 0, 0, 0, 1] + b[0, 1, 0, 0, −2] + c[0, 0, 0, 1, 1], (1 pt. so far) we see that V is the set of linear combinations of three vectors [1, 0, 0, 0, 1], [0, 1, 0, 0, −2] and [0, 0, 0, 1, 1]. That is V = ...
... Sol. Let V denote the set of vectors of the above form. By rewriting [a, b, 0, c, a − 2b + c] = a[1, 0, 0, 0, 1] + b[0, 1, 0, 0, −2] + c[0, 0, 0, 1, 1], (1 pt. so far) we see that V is the set of linear combinations of three vectors [1, 0, 0, 0, 1], [0, 1, 0, 0, −2] and [0, 0, 0, 1, 1]. That is V = ...
Section 6.1 - Gordon State College
... THE UNIT SPHERE The set of all points (vectors) in V that satisfy ||u|| = 1 is called the unit sphere ( or sometimes the unit circle) in V. In R2 and R3 these are points that lie 1 unit, in terms of the inner product, away from the origin. If you are using a different inner product than the dot pro ...
... THE UNIT SPHERE The set of all points (vectors) in V that satisfy ||u|| = 1 is called the unit sphere ( or sometimes the unit circle) in V. In R2 and R3 these are points that lie 1 unit, in terms of the inner product, away from the origin. If you are using a different inner product than the dot pro ...
A.A.12: Multiplication of Powers 2: Multiply and divide monomial
... A.A.12: Multiplication of Powers 2: Multiply and divide monomial expressions with a common base, using the properties of exponents. Note: Use integral exponents only 9 The product of 3x 2 y and 4xy 3 is ...
... A.A.12: Multiplication of Powers 2: Multiply and divide monomial expressions with a common base, using the properties of exponents. Note: Use integral exponents only 9 The product of 3x 2 y and 4xy 3 is ...
MTL101:: Tutorial 3 :: Linear Algebra
... (9) Find a basis B such that [T ]B is a diagonal matrix in case T is diagonalizable. Find P such that [T ]B = P [T ]S P −1 where S is the standard basis in each case. (a) T : C2 → C2 defined by T (x, y) = (y, −x). (b) T : R3 → R3 defined by T (x, y, z) = (5x − 6y − 6z, −x + 4y + 2z, 3x − 6y − 4z). ( ...
... (9) Find a basis B such that [T ]B is a diagonal matrix in case T is diagonalizable. Find P such that [T ]B = P [T ]S P −1 where S is the standard basis in each case. (a) T : C2 → C2 defined by T (x, y) = (y, −x). (b) T : R3 → R3 defined by T (x, y, z) = (5x − 6y − 6z, −x + 4y + 2z, 3x − 6y − 4z). ( ...
R n
... An orthogonal basis is orthonormal if every vector in it has length 1 The standard basis is orthonormal and made up of vectors ei which are all 0's except a 1 at location i ...
... An orthogonal basis is orthonormal if every vector in it has length 1 The standard basis is orthonormal and made up of vectors ei which are all 0's except a 1 at location i ...
Free associative algebras
... studying vector spaces. The theory of bases says that any vector space can be written as a direct sum of lines. In the same way, algebras are a very nice place to study multiplication. There is a notion analogous to direct sum, called tensor product, which makes it possible to multiply things even i ...
... studying vector spaces. The theory of bases says that any vector space can be written as a direct sum of lines. In the same way, algebras are a very nice place to study multiplication. There is a notion analogous to direct sum, called tensor product, which makes it possible to multiply things even i ...
Exercises 5 5.1. Let A be an abelian group. Set A ∗ = HomZ(A,Q/Z
... 5.4. An algebra A over a field K is called a division algebra, if A is a division ring. Give an example of noncommutative division algebra over R. 5.5. Let K be a field, and A a K-linear space with a basis {xi }i∈I . Show that a bilinear map A × A → A, (a, b) 7→ a · b makes A an algebra (not necessa ...
... 5.4. An algebra A over a field K is called a division algebra, if A is a division ring. Give an example of noncommutative division algebra over R. 5.5. Let K be a field, and A a K-linear space with a basis {xi }i∈I . Show that a bilinear map A × A → A, (a, b) 7→ a · b makes A an algebra (not necessa ...
Reducing Dimensionality
... For all x,y belonging to U, we have x+y belongs to U. For all a, we have ax belongs to U. A(ax+by) = aA(x) + bA(y). Geometric interpretation of matrices : Suppose we have a set of points in 2D that form a circle with a1,a2 as basis. Then, matrix multiplication can only rotate the circle and/or scale ...
... For all x,y belonging to U, we have x+y belongs to U. For all a, we have ax belongs to U. A(ax+by) = aA(x) + bA(y). Geometric interpretation of matrices : Suppose we have a set of points in 2D that form a circle with a1,a2 as basis. Then, matrix multiplication can only rotate the circle and/or scale ...
Physics 3730/6720 – Maple 1b – 1 Linear algebra, Eigenvalues and Eigenvectors
... their elements. To do matrix and vector products and see the result you need evalm. (Maple does it, otherwise, but silently.) Multiplication by scalars works with *, but multiplication of matrices with matrices and matrices with vectors requires the special multiplication operator &*. > eigenvals(A) ...
... their elements. To do matrix and vector products and see the result you need evalm. (Maple does it, otherwise, but silently.) Multiplication by scalars works with *, but multiplication of matrices with matrices and matrices with vectors requires the special multiplication operator &*. > eigenvals(A) ...
Chapter 7 Spectral Theory Of Linear Operators In Normed Spaces
... 7.6-1 Definition. An algebra A over a field k is a vector space A over k such that for all x,yA, a unique product xyA is defined with the properties: (1) (xy)z = x(yz) (2) (x+y)z = xz + yz (3) x(y+z) = xy + xz (4) (xy) = (x)y = x(y) for all x,y,zA and scalar k. A is called an algebra with un ...
... 7.6-1 Definition. An algebra A over a field k is a vector space A over k such that for all x,yA, a unique product xyA is defined with the properties: (1) (xy)z = x(yz) (2) (x+y)z = xz + yz (3) x(y+z) = xy + xz (4) (xy) = (x)y = x(y) for all x,y,zA and scalar k. A is called an algebra with un ...
![[S, S] + [S, R] + [R, R]](http://s1.studyres.com/store/data/000054508_1-f301c41d7f093b05a9a803a825ee3342-300x300.png)
Homework 4
... 19) Let V and W be G-modules with characters χ, ψ respectively. Show that χ ⋅ ψ (pointwise product) is the character afforded by the tensor product V ⊗ W. 20) (If you have not seen the ring of algebraic integers – we shall only require the result from c) later on) Let R be an integral domain with qu ...
... 19) Let V and W be G-modules with characters χ, ψ respectively. Show that χ ⋅ ψ (pointwise product) is the character afforded by the tensor product V ⊗ W. 20) (If you have not seen the ring of algebraic integers – we shall only require the result from c) later on) Let R be an integral domain with qu ...
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
... 20.) a.) If U and V are vector spaces over F, and if T is a homomorphism of U onto V with kernel W, then prove that CB ≃ . b.) If V is a finite - dimensional inner product space and if W is a subspace of V, then prove that = ⊕ F . 21.) Prove that every finite - dimensional inner product space has an ...
... 20.) a.) If U and V are vector spaces over F, and if T is a homomorphism of U onto V with kernel W, then prove that CB ≃ . b.) If V is a finite - dimensional inner product space and if W is a subspace of V, then prove that = ⊕ F . 21.) Prove that every finite - dimensional inner product space has an ...
6.837 Linear Algebra Review
... • X, Y, Z is an orthonormalbasis. We can describe any 3D point as a linear combination of these vectors. • How do we express any point as a combination of a new basis U, V, N, given X, Y, Z? 6.837 Linear Algebra Review ...
... • X, Y, Z is an orthonormalbasis. We can describe any 3D point as a linear combination of these vectors. • How do we express any point as a combination of a new basis U, V, N, given X, Y, Z? 6.837 Linear Algebra Review ...
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.