aa1.pdf
... 2 (x + y) ∈ X. An element a ∈ X is said to be an extremal point of X if the only solution (x, y) ∈ X × X of the equation a = 21 (x + y) is x = y = a. Find all extremal points of X. 5. The algebra H of Quaternions is defined as a 4-dimensional R-algebra with basis {1, i, j, k} and the following multi ...
... 2 (x + y) ∈ X. An element a ∈ X is said to be an extremal point of X if the only solution (x, y) ∈ X × X of the equation a = 21 (x + y) is x = y = a. Find all extremal points of X. 5. The algebra H of Quaternions is defined as a 4-dimensional R-algebra with basis {1, i, j, k} and the following multi ...
Vectors - University of Louisville Physics
... There are three forms of multiplication in common use o Multiplication by a scalar If k is a scalar and r is a vector, a = kr is a vector whose magnitude is given by |k||r| with direction either r or -r depending on the sign of k. o ...
... There are three forms of multiplication in common use o Multiplication by a scalar If k is a scalar and r is a vector, a = kr is a vector whose magnitude is given by |k||r| with direction either r or -r depending on the sign of k. o ...
1._SomeBasicMathematics
... n is a semi- (pseudo-) norm if only 1 & 2 hold. A normed vector space is a linear space V endowed with a norm. ...
... n is a semi- (pseudo-) norm if only 1 & 2 hold. A normed vector space is a linear space V endowed with a norm. ...
PDF
... This gives a simple method by which to compute the triple product in a special Jordan algebra. A key instance of the Jordan triple product is the case when x = z (setting x.2 = x.x for notation). Here we get {xyx} = 2(x.y).x − x.2 .y. In a special Jordan algebra this becomes {xyx} = xyx in the assoc ...
... This gives a simple method by which to compute the triple product in a special Jordan algebra. A key instance of the Jordan triple product is the case when x = z (setting x.2 = x.x for notation). Here we get {xyx} = 2(x.y).x − x.2 .y. In a special Jordan algebra this becomes {xyx} = xyx in the assoc ...
Problem set 4
... positions of the particle xk ≡ x(tk ) at equally spaced times t1 , t2 , · · · , tn , with ti+1 − ti = ∆. 1. Assemble the positions of the particle in a column vector with n-components X and display it. < 1 > 2. The velocity is ẋ(t) = lim∆→0 x(t+∆)−x(t) . Define the approximate velocity ẋk at any t ...
... positions of the particle xk ≡ x(tk ) at equally spaced times t1 , t2 , · · · , tn , with ti+1 − ti = ∆. 1. Assemble the positions of the particle in a column vector with n-components X and display it. < 1 > 2. The velocity is ẋ(t) = lim∆→0 x(t+∆)−x(t) . Define the approximate velocity ẋk at any t ...
LINEAR ALGEBRA (1) True or False? (No explanation required
... Explanations: matrices like ( 10 00 ) or ( 11 11 ) are nonzero but do not have an inverse. Matrices have an inverse if and only if they are nonsingular square matrices. If A and B are nonsingular, then so is AB, and its inverse clearly is B −1 A−1 since B −1 A−1 AB = B −1 IB = B −1 B = I. In general ...
... Explanations: matrices like ( 10 00 ) or ( 11 11 ) are nonzero but do not have an inverse. Matrices have an inverse if and only if they are nonsingular square matrices. If A and B are nonsingular, then so is AB, and its inverse clearly is B −1 A−1 since B −1 A−1 AB = B −1 IB = B −1 B = I. In general ...
Lecture 1: Lie algebra cohomology
... particular the de Rham cohomology H• (M) has a well-defined multiplication induced from the wedge product. If M is riemannian, compact and orientable one has the celebrated Hodge decomposition theorem stating that in every de Rham cohomology class there is a unique smooth harmonic form. The second e ...
... particular the de Rham cohomology H• (M) has a well-defined multiplication induced from the wedge product. If M is riemannian, compact and orientable one has the celebrated Hodge decomposition theorem stating that in every de Rham cohomology class there is a unique smooth harmonic form. The second e ...
1 Vector Spaces
... • Def of linear (in)dependence, span, basis. • Examples in F n : – (1, 0, 0, · · · , 0), (0, 1, 0, . . . , 0), . . . , (0, 0, 0, . . . , 1) is a basis for F n – (1, 1, 0), (1, 0, 1), (0, 1, 1) is a basis for F 3 iff F has characteristic 2 • Def: The dimension of a vector space V over F is the size o ...
... • Def of linear (in)dependence, span, basis. • Examples in F n : – (1, 0, 0, · · · , 0), (0, 1, 0, . . . , 0), . . . , (0, 0, 0, . . . , 1) is a basis for F n – (1, 1, 0), (1, 0, 1), (0, 1, 1) is a basis for F 3 iff F has characteristic 2 • Def: The dimension of a vector space V over F is the size o ...
Math 461/561 Week 2 Solutions 1.7 Let L be a Lie algebra. The
... so rearranging gives: [x, [y, z]] − [[x, y], z] = [[z, x], y]. So the left hand side is zero for all x, y, z if and only if [z, x] is in the center, i.e. every commutator lies in Z(L). ...
... so rearranging gives: [x, [y, z]] − [[x, y], z] = [[z, x], y]. So the left hand side is zero for all x, y, z if and only if [z, x] is in the center, i.e. every commutator lies in Z(L). ...
Task 1
... Translate these sentences. 1. There are surfaces which cannot be embedded in three-dimensional Euclidean space without introducing singularities or intersecting itself — these are the unorientable surfaces. _____________________________________________________________________________________________ ...
... Translate these sentences. 1. There are surfaces which cannot be embedded in three-dimensional Euclidean space without introducing singularities or intersecting itself — these are the unorientable surfaces. _____________________________________________________________________________________________ ...
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... Returning to projective geometries, it is known that projective planes can include rather exceptional examples such as planes which do not satisfy Desargues’ theorem (and consequently they do not satisfy Pappaus’ theorem). This means it is not possible to use the standard approach to apply coordinat ...
... Returning to projective geometries, it is known that projective planes can include rather exceptional examples such as planes which do not satisfy Desargues’ theorem (and consequently they do not satisfy Pappaus’ theorem). This means it is not possible to use the standard approach to apply coordinat ...
MULTILINEAR ALGEBRA: THE EXTERIOR PRODUCT This writeup
... Then clearly φ(e1 , · · · , en ) = 1. Also, formal verifications confirm that φ is multilinear and alternating. Thus the mapping property of the nth exterior product gives a commutative diagram ...
... Then clearly φ(e1 , · · · , en ) = 1. Also, formal verifications confirm that φ is multilinear and alternating. Thus the mapping property of the nth exterior product gives a commutative diagram ...
with solutions - MIT Mathematics
... Solution. There are many ways to see that the answer is no for both questions. For example, if both sides are zero, then c can be scaled at will. 7. Consider the (filled) cylinder of radius 2 and height 6 with axis of symmetry along the z-axis. Cut the cylinder in half along the y-z plane and keep ...
... Solution. There are many ways to see that the answer is no for both questions. For example, if both sides are zero, then c can be scaled at will. 7. Consider the (filled) cylinder of radius 2 and height 6 with axis of symmetry along the z-axis. Cut the cylinder in half along the y-z plane and keep ...
Kevin M. Reynolds
... Singular Value Decomposition - Decomposing a matrix into two orthogonal matrices, one being a change of basis in the domain and the other a change of basis for the range such that the matrix becomes diagonal. As an example the method was applied to image compression using MATLAB. ...
... Singular Value Decomposition - Decomposing a matrix into two orthogonal matrices, one being a change of basis in the domain and the other a change of basis for the range such that the matrix becomes diagonal. As an example the method was applied to image compression using MATLAB. ...
Algebras. Derivations. Definition of Lie algebra
... 1.1. Algebras. Let k be a field. An algebra over k (or k-algebra) is a vector space A endowed with a bilinear operation a, b ∈ A 7→ a · b ∈ A. Recall that bilinearity means that for each a ∈ A left and right multiplications by a are linear transformations of vector spaces (i.e. preserve sum and mult ...
... 1.1. Algebras. Let k be a field. An algebra over k (or k-algebra) is a vector space A endowed with a bilinear operation a, b ∈ A 7→ a · b ∈ A. Recall that bilinearity means that for each a ∈ A left and right multiplications by a are linear transformations of vector spaces (i.e. preserve sum and mult ...
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.