211 - SCUM – Society of Calgary Undergraduate Mathematics
... 1. Let T be a linear transformation from R2 → R3 such that • T (1, 1) = (5, −1, 2) • T (0, −2) = (1, 1, −4) Determine T (5, 1). Solution: We must try to express the vector X = (5, 1) as a combination of the vectors Y = (1, 1) and Z = (0, −2), in the form X = aY + bZ. We may do this either by solving ...
... 1. Let T be a linear transformation from R2 → R3 such that • T (1, 1) = (5, −1, 2) • T (0, −2) = (1, 1, −4) Determine T (5, 1). Solution: We must try to express the vector X = (5, 1) as a combination of the vectors Y = (1, 1) and Z = (0, −2), in the form X = aY + bZ. We may do this either by solving ...
Group Theory Summary
... This documents is a summary of Dr. Amihay Hanany’s lectures on Group Theory that I followed during one term between October and December 2008. Some informations also come from the lecture notes form the same course at EPFL and from different sources in books or on the web. Only a few explanations ar ...
... This documents is a summary of Dr. Amihay Hanany’s lectures on Group Theory that I followed during one term between October and December 2008. Some informations also come from the lecture notes form the same course at EPFL and from different sources in books or on the web. Only a few explanations ar ...
COMPUTING RAY CLASS GROUPS, CONDUCTORS AND
... where ψ(α) = cα + a, φ(β) = β and a section σ of φ is given by σ(γ) = aγ + c. (Here denotes the classes in the respective groups, but using the same notation for each will not lead to any confusion as long as we know in which group we are working.) Proof. (1) By assumption, we have a + c = ZK . Thus ...
... where ψ(α) = cα + a, φ(β) = β and a section σ of φ is given by σ(γ) = aγ + c. (Here denotes the classes in the respective groups, but using the same notation for each will not lead to any confusion as long as we know in which group we are working.) Proof. (1) By assumption, we have a + c = ZK . Thus ...
The Past, Present and Future of High Performance Linear Algebra
... • Go back to W and use these b pivot rows • Move them to top, do LU without pivoting • Extra work, but lower order term • Thm: As numerically stable as Partial Pivoting on a larger matrix ...
... • Go back to W and use these b pivot rows • Move them to top, do LU without pivoting • Extra work, but lower order term • Thm: As numerically stable as Partial Pivoting on a larger matrix ...
Polyhedra and Integer Programming
... Exercise 5 asks for a proof of the following corollary. Corollary 1. Let F 1 and F 2 be two inclusion-wise minimal faces of P = {x ∈ Rn : Ax 6 b}, then dim(F 1 ) = dim(F 2 ). We say that a polyhedron contains a line ℓ(x1 , x2 ) with x1 6= x2 ∈ P if ℓ(x1 , x2 ) = {x1 + λ(x2 − x1 ) | λ ∈ R} ⊆ P . A ve ...
... Exercise 5 asks for a proof of the following corollary. Corollary 1. Let F 1 and F 2 be two inclusion-wise minimal faces of P = {x ∈ Rn : Ax 6 b}, then dim(F 1 ) = dim(F 2 ). We say that a polyhedron contains a line ℓ(x1 , x2 ) with x1 6= x2 ∈ P if ℓ(x1 , x2 ) = {x1 + λ(x2 − x1 ) | λ ∈ R} ⊆ P . A ve ...
1.2 row reduction and echelon forms
... 4. The leading entry in each nonzero row is 1. 5. Each leading 1 is the only nonzero entry in its column. An echelon matrix (respectively, reduced echelon matrix) is one that is in echelon form (respectively, reduced echelon form). Property 2 says that the leading entries form an echelon (“steplike” ...
... 4. The leading entry in each nonzero row is 1. 5. Each leading 1 is the only nonzero entry in its column. An echelon matrix (respectively, reduced echelon matrix) is one that is in echelon form (respectively, reduced echelon form). Property 2 says that the leading entries form an echelon (“steplike” ...
Fraction-free matrix factors: new forms for LU and QR factors
... Various applications including robot control and threat analysis have resulted in developing efficient algorithms for the solution of systems of polynomial and differential equations. This involves significant linear algebra subproblems which are not standard numerical linear algebra problems. The “ ...
... Various applications including robot control and threat analysis have resulted in developing efficient algorithms for the solution of systems of polynomial and differential equations. This involves significant linear algebra subproblems which are not standard numerical linear algebra problems. The “ ...
Tranquilli, G.B.; (1965)On the normality of independent random variables implied by intrinsic graph independence without residues."
... These results will be proved without any assumption about the distribution patterns of the marginal r.v.'s Xj ' proVided that these r.v.'s are neither trivial constant r.v.'s nor completely degenerate r.v.'s (still these cases can be considered as limit cases of normal r.v.'s when the variance tends ...
... These results will be proved without any assumption about the distribution patterns of the marginal r.v.'s Xj ' proVided that these r.v.'s are neither trivial constant r.v.'s nor completely degenerate r.v.'s (still these cases can be considered as limit cases of normal r.v.'s when the variance tends ...
Geometry, Statistics, Probability: Variations on a Common Theme
... plane, we have a point x in space. The directed line segment from the origin to x is a vector x. The inner product of any two vectors x and z is still defined by (2.3), for although the vectors are in space, any two of them lie in some plane, and the angle between them is well defined. Thus by addit ...
... plane, we have a point x in space. The directed line segment from the origin to x is a vector x. The inner product of any two vectors x and z is still defined by (2.3), for although the vectors are in space, any two of them lie in some plane, and the angle between them is well defined. Thus by addit ...
Eigenvalue equalities for ordinary and Hadamard products of
... When A and B are real, the two sets of inequalities coniside. However, for complex A and B, as mentioned in [5, p.315], the eigenvalues of AB and AB T may not be the same and they provide different lower bounds in (5) and (6). In Section 5, using a result in Section 4, we determine their equality fo ...
... When A and B are real, the two sets of inequalities coniside. However, for complex A and B, as mentioned in [5, p.315], the eigenvalues of AB and AB T may not be the same and they provide different lower bounds in (5) and (6). In Section 5, using a result in Section 4, we determine their equality fo ...
Random Matrix Theory - Indian Institute of Science
... this model but probabilistically appears even simpler than the previous model as there is more independence. This is a false appearance, but we will come to that later! Patterned random matrices have come into fashion lately. For example, let Xi be i.i.d ...
... this model but probabilistically appears even simpler than the previous model as there is more independence. This is a false appearance, but we will come to that later! Patterned random matrices have come into fashion lately. For example, let Xi be i.i.d ...
