PROBLEM SET ON ANALYSIS 1. Prove or disprove the following
... Assume that there is a positive number k such that ||Ax|| ≥ k||x|| for all x ∈ X. Prove that A(X) is closed in Y . 4. Let T : l2 (Z) → l2 (Z) be the shift operator: T (x)n = xn+1 for any x = (xn ) ∈ l2 (Z). Show that for all λ ∈ C, the operator T − λE is injective and has dense range. 5. Prove that ...
... Assume that there is a positive number k such that ||Ax|| ≥ k||x|| for all x ∈ X. Prove that A(X) is closed in Y . 4. Let T : l2 (Z) → l2 (Z) be the shift operator: T (x)n = xn+1 for any x = (xn ) ∈ l2 (Z). Show that for all λ ∈ C, the operator T − λE is injective and has dense range. 5. Prove that ...
Chapters 1
... Write the equation of a line in standard form that passes through the following points: !8, 3 and 2, ! 2 . ...
... Write the equation of a line in standard form that passes through the following points: !8, 3 and 2, ! 2 . ...
Algebra Wksht 26 - TMW Media Group
... b) Use the graphical exploration you learned in Lesson 25 to show that the system in Problem 1 is consistent. [Solve both equations for y, the number of cake servings, and graph them with the following WINDOW limits: xmin=ymin=0, xmax=125, ymax=250.] 3. The dimension (or size) of a matrix is said to ...
... b) Use the graphical exploration you learned in Lesson 25 to show that the system in Problem 1 is consistent. [Solve both equations for y, the number of cake servings, and graph them with the following WINDOW limits: xmin=ymin=0, xmax=125, ymax=250.] 3. The dimension (or size) of a matrix is said to ...
Quantum Computation
... Representing a Qubit A qubit |ψi = α |0i + β |1i with |α|2 + |β|2 = 1 can be represented: |ψi = cos(θ/2) |0i + e iϕ sin(θ/2) |1i , where θ ∈ [0, π] and ϕ ∈ [0, 2π]. Using polar coordinates we have: |ψi = r0 e iφ0 |0i + r1 e iφ1 |1i , with r02 + r12 = 1. Take r0 = cos(ρ) and r1 = sin(ρ) for some ρ. S ...
... Representing a Qubit A qubit |ψi = α |0i + β |1i with |α|2 + |β|2 = 1 can be represented: |ψi = cos(θ/2) |0i + e iϕ sin(θ/2) |1i , where θ ∈ [0, π] and ϕ ∈ [0, 2π]. Using polar coordinates we have: |ψi = r0 e iφ0 |0i + r1 e iφ1 |1i , with r02 + r12 = 1. Take r0 = cos(ρ) and r1 = sin(ρ) for some ρ. S ...
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.