Document
... Given scaling vector S Take the matrix A Scale A’s x axis’ x component by S0 Scale A’s y axis’ y component by S1 Scale A’s z axis’ z component by S2 ...
... Given scaling vector S Take the matrix A Scale A’s x axis’ x component by S0 Scale A’s y axis’ y component by S1 Scale A’s z axis’ z component by S2 ...
5. Geometry of numbers
... Minkowski on the existence of lattice points in symmetric convex bodies given in 5.1. Let V be a vector space of finite dimension n over the field R of real numbers, and h·, ·i : V × V → R a scalar product, i.e. a positive definite bilinear form on V × V . The scalar product induces a notion of volu ...
... Minkowski on the existence of lattice points in symmetric convex bodies given in 5.1. Let V be a vector space of finite dimension n over the field R of real numbers, and h·, ·i : V × V → R a scalar product, i.e. a positive definite bilinear form on V × V . The scalar product induces a notion of volu ...
A Few Words on Spaces, Vectors, and Functions
... The vectors form an Abelian group because x + y = y + x , the unit (“neutral”) element is (0, 0, . . . 0), the inverse element to (a1 , a2 , . . . a N ) is equal to (−a1 , −a2 , . . . −a N ). Thus, the vectors form a group. “Multiplication” of a vector by a real number α means α(a1 , a2 , . . . a N ...
... The vectors form an Abelian group because x + y = y + x , the unit (“neutral”) element is (0, 0, . . . 0), the inverse element to (a1 , a2 , . . . a N ) is equal to (−a1 , −a2 , . . . −a N ). Thus, the vectors form a group. “Multiplication” of a vector by a real number α means α(a1 , a2 , . . . a N ...
aa3.pdf
... 2. Fix a ring A and an idempotent e ∈ A. The subset eAe ⊂ A is a ring with unit e (thus, eAe is not a subring of A, according to our definitions). Prove the following: (i) For any A-module M , there is a natural eAe-module structure on the subgroup eM ⊂ M (here eM = {em | m ∈ M }). Furthermore, for ...
... 2. Fix a ring A and an idempotent e ∈ A. The subset eAe ⊂ A is a ring with unit e (thus, eAe is not a subring of A, according to our definitions). Prove the following: (i) For any A-module M , there is a natural eAe-module structure on the subgroup eM ⊂ M (here eM = {em | m ∈ M }). Furthermore, for ...
11-25 PPT
... 2. Kyle has 5 more than one fourth as many Legos as Tom. 3. Moesha’s music library has 17 more than 2 times the songs as Damian’s. 4. Ciera has three more the one half the number of purses as Aisha. ...
... 2. Kyle has 5 more than one fourth as many Legos as Tom. 3. Moesha’s music library has 17 more than 2 times the songs as Damian’s. 4. Ciera has three more the one half the number of purses as Aisha. ...
Reformulated as: either all Mx = b are solvable, or Mx = 0 has
... Definition 9. A linear transformation T : U ! V which is onto to one and onto is called an isomorphism of vector spaces, and U and V are called isomorphic vector spaces. Whenever two vector spaces are isomorphic2 and T : U ! V is an isomorphism, then any property of U that can be written using the v ...
... Definition 9. A linear transformation T : U ! V which is onto to one and onto is called an isomorphism of vector spaces, and U and V are called isomorphic vector spaces. Whenever two vector spaces are isomorphic2 and T : U ! V is an isomorphism, then any property of U that can be written using the v ...
notes
... What are the implications of the different ways of organizing matrix multiplication? Let’s compare the time taken to run the different versions of the multiply on a pair of square matrices of dimension 510 through 515. ...
... What are the implications of the different ways of organizing matrix multiplication? Let’s compare the time taken to run the different versions of the multiply on a pair of square matrices of dimension 510 through 515. ...
24. Orthogonal Complements and Gram-Schmidt
... exercise that U ∩ V is a subspace of W , and that U ∪ V was not a subspace. However, span U ∪ V is a subspace2 . Notice that all elements of span U ∪ V take the form u + v with u ∈ U and v ∈ V . When U ∩ V = {0W }, we call the subspace span U ∪ V the direct sum of U and V , written: U ⊕ V = span U ∪ ...
... exercise that U ∩ V is a subspace of W , and that U ∪ V was not a subspace. However, span U ∪ V is a subspace2 . Notice that all elements of span U ∪ V take the form u + v with u ∈ U and v ∈ V . When U ∩ V = {0W }, we call the subspace span U ∪ V the direct sum of U and V , written: U ⊕ V = span U ∪ ...
Proposition 7.3 If α : V → V is self-adjoint, then 1) Every eigenvalue
... If V is an inner product space, then any subspace W is also an inner product space with respect to the same inner product. Furthermore, if α : V → V is self-adjoint and W is α-invariant, then α|W is self-adjoint as well. Lemma 7.5 If α : V → V is self-adjoint and W is α-invariant, then W ⊥ = {v ∈ V ...
... If V is an inner product space, then any subspace W is also an inner product space with respect to the same inner product. Furthermore, if α : V → V is self-adjoint and W is α-invariant, then α|W is self-adjoint as well. Lemma 7.5 If α : V → V is self-adjoint and W is α-invariant, then W ⊥ = {v ∈ V ...
![Lecture 2A [pdf]](http://s1.studyres.com/store/data/008845380_1-be6e705eb191ba98899bd42e027ab326-300x300.png)
Lecture 2A [pdf]
... To an ant walking on M , a geodesic curve is like a straight line in the plane. The geodesics on a sphere are the great circles. ...
... To an ant walking on M , a geodesic curve is like a straight line in the plane. The geodesics on a sphere are the great circles. ...
course outline - Clackamas Community College
... Apply the determinant to find areas of parallelograms and volumes of parallelepipeds. Apply the determinant to find the area or volume of a linear transformation of an area or volume. Use row reduction to find the eigenspace and eigenvectors of a square matrix for a given real eigenvalue. Find the r ...
... Apply the determinant to find areas of parallelograms and volumes of parallelepipeds. Apply the determinant to find the area or volume of a linear transformation of an area or volume. Use row reduction to find the eigenspace and eigenvectors of a square matrix for a given real eigenvalue. Find the r ...
vector - e-CTLT
... • Zero or Null Vector: A vector whose initial and terminal points are coincident is called zero or null vector. It is denoted by 0. • Unit Vector: A vector whose magnitude is unity is called a unit vector in direction of a which is denoted by a i.e a =a/|a| • Collinear or Parallel Vectors: Two or mo ...
... • Zero or Null Vector: A vector whose initial and terminal points are coincident is called zero or null vector. It is denoted by 0. • Unit Vector: A vector whose magnitude is unity is called a unit vector in direction of a which is denoted by a i.e a =a/|a| • Collinear or Parallel Vectors: Two or mo ...
Summary of week 6 (lectures 16, 17 and 18) Every complex number
... (where (u1 , u2 , . . . , un ) is any orthogonal basis for U ) that P is a linear map. We can use orthogonal projections to show that every finite-dimensional inner product space has an orthogonal basis. More generally, suppose that V is an inner product space and U1 ⊂ U2 ⊂ · · · Ud is an increasing ...
... (where (u1 , u2 , . . . , un ) is any orthogonal basis for U ) that P is a linear map. We can use orthogonal projections to show that every finite-dimensional inner product space has an orthogonal basis. More generally, suppose that V is an inner product space and U1 ⊂ U2 ⊂ · · · Ud is an increasing ...
Vector spaces, norms, singular values
... You have had a previous class in which you learned the basics of linear algebra, and you will have plenty of practice with these concepts over the semester. This brief refresher lecture is supposed to remind you of what you've already learned and introduce a few things you may not have seen. It also ...
... You have had a previous class in which you learned the basics of linear algebra, and you will have plenty of practice with these concepts over the semester. This brief refresher lecture is supposed to remind you of what you've already learned and introduce a few things you may not have seen. It also ...
Lec 25: Coordinates and Isomorphisms. [Here should be an
... Theorem. Vector spaces of different dimensions are not isomorphic. Exercise: prove this theorem (hint: show first that any isomorphism takes a basis to a basis). For example, spaces Pol(5) and Mat(3, 3) are not isomorphic and Pol(5), Mat(2, 3) are. Work out conditions for the space Pol(n) and Pol(k, ...
... Theorem. Vector spaces of different dimensions are not isomorphic. Exercise: prove this theorem (hint: show first that any isomorphism takes a basis to a basis). For example, spaces Pol(5) and Mat(3, 3) are not isomorphic and Pol(5), Mat(2, 3) are. Work out conditions for the space Pol(n) and Pol(k, ...
MATH 323.502 Exam 2 Solutions April 14, 2015 1. For each
... (c) For a 2 × 3 matrix A, the dimension of N (A) can be 1, 2 or 3, but not 0. (d) For a 3 × 5 matrix C, if for every b ∈ R3 the system Cx = b is consistent, then the rank of C is 3. (e) Suppose that v1 , v2 , v3 are linearly independent vectors in R6 . Suppose that w1 , w2 , w3 are linearly independ ...
... (c) For a 2 × 3 matrix A, the dimension of N (A) can be 1, 2 or 3, but not 0. (d) For a 3 × 5 matrix C, if for every b ∈ R3 the system Cx = b is consistent, then the rank of C is 3. (e) Suppose that v1 , v2 , v3 are linearly independent vectors in R6 . Suppose that w1 , w2 , w3 are linearly independ ...
What can the answer be? II. Reciprocal basis and dual vectors
... and c . [This statement is a bit loose and glosses over certain technical details, but is quite acceptable at the present level of rigour.] It turns out that we can prove that the dual space is actually isomorphic to the original space, provided the latter is finite-dimensional (in our case, it is ...
... and c . [This statement is a bit loose and glosses over certain technical details, but is quite acceptable at the present level of rigour.] It turns out that we can prove that the dual space is actually isomorphic to the original space, provided the latter is finite-dimensional (in our case, it is ...
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.