Solutions - NIU Math
... 6. (a) Let α be a fixed element of Sn . Show that φα : Sn → Sn defined by φα (σ) = ασα−1 , for all σ ∈ Sn , is a one-to-one and onto function. Solution: If φα (σ) = φα (τ ), for σ, τ ∈ Sn , then ασα−1 = ατ α−1 . We can multiply on the left by α−1 and on the right by α, to get σ = τ , so φα is one-to ...
... 6. (a) Let α be a fixed element of Sn . Show that φα : Sn → Sn defined by φα (σ) = ασα−1 , for all σ ∈ Sn , is a one-to-one and onto function. Solution: If φα (σ) = φα (τ ), for σ, τ ∈ Sn , then ασα−1 = ατ α−1 . We can multiply on the left by α−1 and on the right by α, to get σ = τ , so φα is one-to ...
Joint Regression and Linear Combination of Time
... Note that the coefficient matrix not only contains the original polynomials f1 , . . . , fs but also ’shifted’ versions where we define a shift as a multiplication with a monomial. The dependence of this matrix on the degree d is of crucial importance, hence the notation M (d). It can be shown [8] t ...
... Note that the coefficient matrix not only contains the original polynomials f1 , . . . , fs but also ’shifted’ versions where we define a shift as a multiplication with a monomial. The dependence of this matrix on the degree d is of crucial importance, hence the notation M (d). It can be shown [8] t ...
BCPS Intensified Algebra Final Exam 2013/2014
... 7. The box office took in a total of $2905 in paid admissions for the high-school musical. Adult tickets cost $8 each, and student tickets cost $3 each. If 560 people attended the show, how many were students? Let s = the number of students attending, and let a = the number of adults attending. Whi ...
... 7. The box office took in a total of $2905 in paid admissions for the high-school musical. Adult tickets cost $8 each, and student tickets cost $3 each. If 560 people attended the show, how many were students? Let s = the number of students attending, and let a = the number of adults attending. Whi ...
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.