HOMEWORK 3, due December 15 1. Adjoint operators. Let H be a
... Show that spectrum of A∗ is the complex conjugation of the spectrum of A. An operator is called selfadjoint if A∗ = A. When is a product of two self-adjoint operators self-adjoint? What do we know about the spectrum of a self-adjoint operator? 2. Hellinger-Toeplitz theorem. Let A : H → H be a linear ...
... Show that spectrum of A∗ is the complex conjugation of the spectrum of A. An operator is called selfadjoint if A∗ = A. When is a product of two self-adjoint operators self-adjoint? What do we know about the spectrum of a self-adjoint operator? 2. Hellinger-Toeplitz theorem. Let A : H → H be a linear ...
Math 215 HW #4 Solutions
... in the plane. Since this matrix clearly has rank 1, we know that the dimension of the nullspace is 4 − 1 = 3, so the plane x + 2y − 3z − t = 0, which is the same as the nullspace, is also three-dimensional and so cannot contain four linearly independent vectors) 3. Problem 2.3.26. Suppose S is a fiv ...
... in the plane. Since this matrix clearly has rank 1, we know that the dimension of the nullspace is 4 − 1 = 3, so the plane x + 2y − 3z − t = 0, which is the same as the nullspace, is also three-dimensional and so cannot contain four linearly independent vectors) 3. Problem 2.3.26. Suppose S is a fiv ...
Linear Algebra
... Matrix multiplication AB: apply transformation B first, and then again transform using A! Heads up: multiplication is NOT commutative! Note: If A and B both represent either pure “rotation” or “scaling” they can be interchanged (i.e. AB = BA) Shivkumar Kalyanaraman ...
... Matrix multiplication AB: apply transformation B first, and then again transform using A! Heads up: multiplication is NOT commutative! Note: If A and B both represent either pure “rotation” or “scaling” they can be interchanged (i.e. AB = BA) Shivkumar Kalyanaraman ...
Part I - Penn Math - University of Pennsylvania
... same element [Z2 ] + [Z3 ], though Z6 and S3 are not isomorphic. Exercise 2 Express in terms of generators the following elements of G: a) [Zn , n is not a prime. b) [Sn ] where Sn is the group of permutations of n objects. c) [Tn (Fq )] where Tn (Fq ) is the group of all invertible upper triangular ...
... same element [Z2 ] + [Z3 ], though Z6 and S3 are not isomorphic. Exercise 2 Express in terms of generators the following elements of G: a) [Zn , n is not a prime. b) [Sn ] where Sn is the group of permutations of n objects. c) [Tn (Fq )] where Tn (Fq ) is the group of all invertible upper triangular ...
Isolated points, duality and residues
... evaluations at points play an important role. We would like to pursue this idea further and to study evaluations by themselves. In other words, we are interested here, by the properties of the dual of the algebra of polynomials. Our objective is to understand the properties of the dual, which can b ...
... evaluations at points play an important role. We would like to pursue this idea further and to study evaluations by themselves. In other words, we are interested here, by the properties of the dual of the algebra of polynomials. Our objective is to understand the properties of the dual, which can b ...
Atom structures
... isomorphic copy of it. This is easily seen by a simple cardinality argument: for any countably infinite atomic algebra A, the algebra (At A)+ will be uncountable. This does not indicate however, that in its own right, the construction of taking the atom structure of an arbitrary atomic bao has recei ...
... isomorphic copy of it. This is easily seen by a simple cardinality argument: for any countably infinite atomic algebra A, the algebra (At A)+ will be uncountable. This does not indicate however, that in its own right, the construction of taking the atom structure of an arbitrary atomic bao has recei ...
Compositions of Linear Transformations
... First, note for any ~x, ~y ∈ R2 , we have S ◦ T (~x + ~y ) = S(T (~x + ~y )) = S(T (~x)) + S(T (~y )) = S ◦ T (~x) + S ◦ T (~y ). Here, the second and third equal signs come from the linearity of T and S, respectively. Next, note that for any ~x ∈ R2 and any scalar k, we have S ◦ T (k~x) = S(T (k~x) ...
... First, note for any ~x, ~y ∈ R2 , we have S ◦ T (~x + ~y ) = S(T (~x + ~y )) = S(T (~x)) + S(T (~y )) = S ◦ T (~x) + S ◦ T (~y ). Here, the second and third equal signs come from the linearity of T and S, respectively. Next, note that for any ~x ∈ R2 and any scalar k, we have S ◦ T (k~x) = S(T (k~x) ...
NONCOMMUTATIVE JORDAN ALGEBRAS OF
... case by the fact that, for an absolutely primitive idempotent u in a general (commutative) Jordan algebra A of characteristic p>0, the structure of Au(l)—the subalgebra on which u acts as an identity—is not known. When this result is known, it may yield not only a determination of (commutative) Jord ...
... case by the fact that, for an absolutely primitive idempotent u in a general (commutative) Jordan algebra A of characteristic p>0, the structure of Au(l)—the subalgebra on which u acts as an identity—is not known. When this result is known, it may yield not only a determination of (commutative) Jord ...
Linear Algebra and Matrices
... Determinants • Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations (i.e. GLMs). • The determinant is a function that associates a scalar det(A) to every square matrix A. – Input is nxn matrix – Output is a single number (real or com ...
... Determinants • Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations (i.e. GLMs). • The determinant is a function that associates a scalar det(A) to every square matrix A. – Input is nxn matrix – Output is a single number (real or com ...
The matrix of a linear operator in a pair of ordered bases∗
... Example 1. Let us give some examples of a linear operator A : V → W : a) V = W = R2 , A(x1 , x2 ) = (x1 , −x2 ) (reflection of a plane in the x1 - axis); b) V = W = R2 , A(x1 , x2 ) = (−x1 , −x2 ) (symmetry of a plane about the origin); c) V = W = R2 , A(x1 , x2 ) = (x1 , 0) (orthogonal projection o ...
... Example 1. Let us give some examples of a linear operator A : V → W : a) V = W = R2 , A(x1 , x2 ) = (x1 , −x2 ) (reflection of a plane in the x1 - axis); b) V = W = R2 , A(x1 , x2 ) = (−x1 , −x2 ) (symmetry of a plane about the origin); c) V = W = R2 , A(x1 , x2 ) = (x1 , 0) (orthogonal projection o ...
Math 110, Fall 2012, Sections 109-110 Worksheet 121 1. Let V be a
... If {xi } is any orthonormal basis of eigenvectors of A, then each xi is also an eigenvector of S by construction, so S possess an orthonormal basis of eigenvectors. Thus S is normal, and by construction has all nonnegative eigenvalues, so S is positive semidefinite. We have S 2 (v1 + · · · + vk ) = ...
... If {xi } is any orthonormal basis of eigenvectors of A, then each xi is also an eigenvector of S by construction, so S possess an orthonormal basis of eigenvectors. Thus S is normal, and by construction has all nonnegative eigenvalues, so S is positive semidefinite. We have S 2 (v1 + · · · + vk ) = ...
Q(xy) = Q(x)Q(y).
... IV. C3($) = C(C2(4>)),a Cayley algebra with basis {1, i, j, ft, /, il, jl, kl} where 1= -I, l2= vl (y^O). Since C3 is not associative the Cayley-Dickson construction ends here. It can be shown [l] that every composition algebra over <£>is isomorphic to one of the above algebras (for suitable choice ...
... IV. C3($) = C(C2(4>)),a Cayley algebra with basis {1, i, j, ft, /, il, jl, kl} where 1= -I, l2= vl (y^O). Since C3 is not associative the Cayley-Dickson construction ends here. It can be shown [l] that every composition algebra over <£>is isomorphic to one of the above algebras (for suitable choice ...
Slides
... associated free object in C . The classical notions can be recovered by replacing Set. Definition ([3, 4]) A normed set is a pair (S, f ), where S is a set and f a function from S to [0, ∞). Given two normed sets (S, f ) and (T , g ), a function φ : S → T is contractive if g (φ(s)) ≤ f (s) for all s ...
... associated free object in C . The classical notions can be recovered by replacing Set. Definition ([3, 4]) A normed set is a pair (S, f ), where S is a set and f a function from S to [0, ∞). Given two normed sets (S, f ) and (T , g ), a function φ : S → T is contractive if g (φ(s)) ≤ f (s) for all s ...
Gates accept concurrent behavior
... where a choice is made based on the information accumulated in the current state. We can have a choice between several states, like a case statement in C. Dilemma: v A w = 0 and # ( v V w ) expresses no basis for choosing between ZI and w , since it means making the choice when no events have occurr ...
... where a choice is made based on the information accumulated in the current state. We can have a choice between several states, like a case statement in C. Dilemma: v A w = 0 and # ( v V w ) expresses no basis for choosing between ZI and w , since it means making the choice when no events have occurr ...
THE COHOMOLOGY RING OF FREE LOOP SPACES 1. Introduction
... The algebra H ∗ (X S ) can sometimes be computed via spectral sequences. For example, Kuribayashi and Yamaguchi [27], by solving extensions problems by applications of the Steenrod operations, were able to compute, via the Eilenberg-Moore spectral sequence, the alge1 bra H ∗ (X S ) for some simple s ...
... The algebra H ∗ (X S ) can sometimes be computed via spectral sequences. For example, Kuribayashi and Yamaguchi [27], by solving extensions problems by applications of the Steenrod operations, were able to compute, via the Eilenberg-Moore spectral sequence, the alge1 bra H ∗ (X S ) for some simple s ...
Semidirect Products - Mathematical Association of America
... H(X, 1), obtained from the proof of Cayley's theorem, is called the left regular representation of X in Sym(X). In the case of the additive group R+, the set of translations {cIl,b} = {x - x + b} is the regular representationof IR+. Notice that the left regular representation of X, H(X, 1), is a nor ...
... H(X, 1), obtained from the proof of Cayley's theorem, is called the left regular representation of X in Sym(X). In the case of the additive group R+, the set of translations {cIl,b} = {x - x + b} is the regular representationof IR+. Notice that the left regular representation of X, H(X, 1), is a nor ...
alg6.1
... 6-1 Integer Exponents Check It Out! Example 3b Evaluate the expression for the given values of the variables. for a = –2 and b = 6 Substitute –2 for a and 6 for b. Simplify expressions with exponents. Write the power in the denominator as a product. Simplify the denominator. ...
... 6-1 Integer Exponents Check It Out! Example 3b Evaluate the expression for the given values of the variables. for a = –2 and b = 6 Substitute –2 for a and 6 for b. Simplify expressions with exponents. Write the power in the denominator as a product. Simplify the denominator. ...
INTRODUCTION TO RATIONAL CHEREDNIK ALGEBRAS
... 1.2. Invariants of complex reflection groups. The ring of functions C[V ] admits a representation of W . Its W -invariants admit the following description. Theorem 1.1 (Shephard-Todd, Chevalley). The algebra of invariants C[V ]W is a polynomial ring generated by homogeneous elements of degree d1 , . ...
... 1.2. Invariants of complex reflection groups. The ring of functions C[V ] admits a representation of W . Its W -invariants admit the following description. Theorem 1.1 (Shephard-Todd, Chevalley). The algebra of invariants C[V ]W is a polynomial ring generated by homogeneous elements of degree d1 , . ...
Vectors and Vector Spaces
... Remark 1.4.1. In the proof it was important to begin with any spanning set for V and to extract vectors from it as we did. Assuming merely that there exists a finite spanning set and extracting vectors directly from V leads to a problem of terminus. That is, when can we say that the new linearly ind ...
... Remark 1.4.1. In the proof it was important to begin with any spanning set for V and to extract vectors from it as we did. Assuming merely that there exists a finite spanning set and extracting vectors directly from V leads to a problem of terminus. That is, when can we say that the new linearly ind ...
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.