A Brief Primer on Matrix Algebra
... Matrix inversion is both difficult and tedious to do with only a calculator, so one of the major advantages of having statistical software is that the computer can do the heavy lifting. Despite not having to calculate an inverted matrix by hand, several of their features are important. First, the in ...
... Matrix inversion is both difficult and tedious to do with only a calculator, so one of the major advantages of having statistical software is that the computer can do the heavy lifting. Despite not having to calculate an inverted matrix by hand, several of their features are important. First, the in ...
Elements of Convex Optimization Theory
... (c) By the linearity of the inner product, x = x B implies (x j B 0 ) = x (B j B 0 ) : By part (b), (B j B 0 ) is invertible, and the claimed expression for x follows. Two vectors x and y are orthogonal if (x j y) = 0. A set of vectors is orthogonal if its elements are pairwise orthogonal. A basis B ...
... (c) By the linearity of the inner product, x = x B implies (x j B 0 ) = x (B j B 0 ) : By part (b), (B j B 0 ) is invertible, and the claimed expression for x follows. Two vectors x and y are orthogonal if (x j y) = 0. A set of vectors is orthogonal if its elements are pairwise orthogonal. A basis B ...
Reed-Muller codes
... Synopsis. The minimum distance of a perfect code cannot exceed 7 unless the code is a repetition code. This is disappointingly low. In this final section of the course, we construct Reed-Muller codes, a family of codes with large minimum distance. Unfortunately, they are not perfect. First, we cover ...
... Synopsis. The minimum distance of a perfect code cannot exceed 7 unless the code is a repetition code. This is disappointingly low. In this final section of the course, we construct Reed-Muller codes, a family of codes with large minimum distance. Unfortunately, they are not perfect. First, we cover ...
q-linear functions, functions with dense graph, and everywhere
... In this section, we construct an infinite dimensional vector space whose nonzero elements are everywhere surjective Q-linear functions. We will start by characterizing this kind of functions. Clearly, from the above definition, a function f : R −→ R is everywhere surjective if and only if f −1 (t) i ...
... In this section, we construct an infinite dimensional vector space whose nonzero elements are everywhere surjective Q-linear functions. We will start by characterizing this kind of functions. Clearly, from the above definition, a function f : R −→ R is everywhere surjective if and only if f −1 (t) i ...
Conservative vector fields
... force is somewhat peculiar, it must have some additional properties with respect to a generic vector field. The purpose of these notes is to explain very briefly what are the characteristics that differentiate the gravitational field from a generic vector field to which it is not possible to associa ...
... force is somewhat peculiar, it must have some additional properties with respect to a generic vector field. The purpose of these notes is to explain very briefly what are the characteristics that differentiate the gravitational field from a generic vector field to which it is not possible to associa ...
mathematics 217 notes
... The characteristic polynomial of an n×n matrix A is the polynomial χA (λ) = det(λI −A), a monic polynomial of degree n; a monic polynomial in the variable λ is just a polynomial with leading term λn . Note that similar matrices have the same characteristic polynomial, since det(λI − C −1 AC) = det C ...
... The characteristic polynomial of an n×n matrix A is the polynomial χA (λ) = det(λI −A), a monic polynomial of degree n; a monic polynomial in the variable λ is just a polynomial with leading term λn . Note that similar matrices have the same characteristic polynomial, since det(λI − C −1 AC) = det C ...
EXAMPLE 6 Find the gradient vector field of . Plot the gradient vector
... gradient vectors are perpendicular to the level curves, as we would expect from Section 11.6. Notice also that the gradient vectors are long where the level curves are close to each other and short where they are farther apart. That’s because the length of the gradient vector is the value of the dir ...
... gradient vectors are perpendicular to the level curves, as we would expect from Section 11.6. Notice also that the gradient vectors are long where the level curves are close to each other and short where they are farther apart. That’s because the length of the gradient vector is the value of the dir ...
Transformation of the Navier-Stokes Equations in Curvilinear
... form of these equations and their derivation in tensor calculus textbooks [1]–[3]. However, its have not been used widely in numerical simulations, because of the calculation of the covariant derivatives in curvilinear coordinate systems is generally very complicate and spent time to much for calcul ...
... form of these equations and their derivation in tensor calculus textbooks [1]–[3]. However, its have not been used widely in numerical simulations, because of the calculation of the covariant derivatives in curvilinear coordinate systems is generally very complicate and spent time to much for calcul ...
1 VECTOR SPACES AND SUBSPACES
... • A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V . In general, all ten vector space axioms must be verified to show that a set W with addition and scalar multiplication forms a vector space. However, if ...
... • A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V . In general, all ten vector space axioms must be verified to show that a set W with addition and scalar multiplication forms a vector space. However, if ...
Measurement and Uncertainties
... An important skill for physicists is to understand whether or not the physics being considered is sensible. When performing a calculation in which someone’s mass was calculated to be 5000 kg, this should ring alarm bells. Since average adult masses (“weights”) will usually be 60–90 kg, a value of 50 ...
... An important skill for physicists is to understand whether or not the physics being considered is sensible. When performing a calculation in which someone’s mass was calculated to be 5000 kg, this should ring alarm bells. Since average adult masses (“weights”) will usually be 60–90 kg, a value of 50 ...
