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Linear Algebra

Differential Operators: Eigenvalues and Eigenfunctions

Please see the attached file for the fully formatted problems. Let L = with boundary conditions u(0) = 0, u'(O) = u(1) ,so that the domain of L is S = {u Lu is square integrable; u(0) = 0, u'(O) = u(1)}. (a) For the above differential operator FIND S* for the adjoint with respect to (v,u) =S 1-->0 v-bar u dx and compare S

Eignevalues and Eigenvectors: Example Problem

Please see the attached file for the fully formatted problems. The Fourier transform, call it F, is a linear one-to-one operator from the space of square-integrable functions onto itself. (In fact, we also know that F is an "isometric" mapping, but we will not need this feature in this problem). Indeed, Note that here x an

Linear algebra

Determine the characteristic values of the given matrix and find the corresponding vectors: [ 2 -2 1 ] [ 1 -1 1 ] [ -3 2 -2 ]

Linear Algebra : Wronskian

Compute the Wronskian of the given set of functions, then determine whether the function is linearly dependent or linearly independent: x^2 - x, x^2 + x, x^2, all x

Linear Algebra And Differential Equations: Real Vector Space

Determine if the given set constitutes a real vector space. The operations of "multiplication by a number" and "addition" are understood to be the usual operations associated with the elements of the set: The set of all elements of R^3 with first component 0

Change of Basis: Eigenvectors

For the problem, refer to the linear transformation T: R^3 --> R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). Write the change of basis matrix K from the basis F of R^3 which consists of the eigenvectors of T to the standard basis E for R^3.

Ellipsoid to canonical form

Problem attached. (a) Find the shortest and the largest distance from the origin to the surface of the ellipsoid. (b) Find the principal axes of the ellipsoid.

Eigenvector Estimation : Inverse Power Method

Please see the attached file for the fully formatted problems. Use the inverse power method to estimate the eigenvector corresponding to the eigenvalue with smallest absolute value for the matrix -1 -2 -1 A= -2 -4 -3 2 2 1 where X0= [1,1,-1]. In finding A-1 use exact arithmetic with fractions. ln applyi

Cosets Groups

Note: C means set containment (not proper) |G:H| means index of subgroup H in G U means union of sets E means belonging to Let K C H C G be groups. Show that both |G:H| and |H:K| are finite if and only if |G:K| is finite, and then |G:K| = |G:H||H:K|. Hint: if |H:K| = n, let Kh1, Kh2, ..., Khn be the distinct cosets of

Linear Optimization Applications

Please see the attached file for full problem description. The local drug store sells a wide variety of cold medications. During the particularly harsh winter, the only three types left on the shelf were in the children's section. These are:

Signals - System Properties

I have difficulty in determining whether the signals are memoryless or causal. Please see the attached file for full problem description.

Linear velocity

Find the linear velocity of a point on the edge of a drum rotating 52 times per minute. The diameter of the wheel is 16.0in. Please show me all the steps thank you

Proof : Diagonalization of Matrices

Please see the attached file for full problem description. Write a proof for the following statement: If A is an n x n upper triangular matrix with no two diagonal elements the same, then A is similar to a diagonal matrix. Show work.

Matrices : Finding the Rank

How to find rank of a matrix: definitions and an example (4*4 matrix) with detailed explanations. Find the rank of A= [1 0 2 0] [4 0 3 0] [5 0 -1 0] [2 -3 1 1]. Show all work.

Linear algebra problems

I have two questions that I need help with. 1) How would you find a basis of the kernel, a basis of the image and determine the dimension of each for this matrix? The matrix is in the attachment. 2) Are the following 3 vectors linearly dependent? (see attachment for the three vectors) How can you decide? I hope y

Linear functions and equations

1. Determine whether each of the following is a function or not. (a) f(x) = 1 if x>1 = 0 otherwise (b) f(x) = 2 if x>0 = -2 if x<0 = 2 or -2 if x = 0 = 0 otherwise (c) f(x) = 5/x 2. Suppose you have a lemonade stand, and when you charge $1 per cup of lemonade you se

Systems of Linear Equations : 3 Unknowns (Echelon Method)

The problem is to find all the possible solutions to the following: Eq 1: x + y = 2 Eq 2: y + z = 3 Eq 3: x + 2y + z = 5 I set up my matricies in the following: 1 1 0 2 0 1 1 3 1 2 1 5 operation 1: (-1*row 1 +row 3) 1 1 0 2 0 1 1 3 0 1 1 3 operation 2: (-1*row 2 +row 3) 1 1 0 2 0 1 1 3 0 0

Generating Linear Algebra

Vector Space and Subspaces Euclidian 3-space Problem:- Show that the vectors u1 = (1,2,3), u2 = (0,1,2), u3 = (2,0,1) generate R3(R).

Euclidean Linear Dependence

1. Is H= the column vectors of the matrix M= [1 0 2 0] cR^4 linearly independent or [4 0 3 0] linearly dependent? [5 0 -1 0] Justify your answer.

Linear algebra

Suppose S is a linear space defined below. Are the following mappings L linear transformations from S into itself? If answer is yes, find the matrix representations of formations (in standard basis): (a) S=P4, L(p(x))=p(0)+x*p(1)+x^2*p(2)+X^3*p(4) (b) S=P4, L(p(x))=x^3+x*p'(x)+p(0) (c) S is a subspace of C[0,1] formed by

Solutions for equations

Consider the equation Ax=b, with a=(1 1 a) (1 -1 1) (2 -1 -1) b=(6+b) ( b ) ( b ) for which values of a,b this system has no solutions? infinitely many solutions? unique solution? if possible, find the solution x explicitly in terms of a,b.

Solutions of Linear Equations

I would like a short explanation of Gaussian Elimination with partial pivoting and Gauss-Seidel. Also, explain when each applies or when one is better than the other. Please include some examples.

Wave equation with mixed boundary conditions

Uxx means second derivative with respect to x Uyy means second derivative with respect to y Uxx + Uyy = 0, 0 < x < pi, 0 < y < 1 Ux(0,y) = 0 = U(pi,y), 0 < y < 1 U(x,0) = 1, U(x,1) = 0, 0 < x < pi Please show all work including how eigenvalues and eigenvectors are derived. Thank you

Linear Algebra: Vectors - Inner Product

Show that the functions x and x^2 are orthogonal in P5 with inner product defined by ( <p,q>=sum from i=1 to n of p(xi)*q*(xi) ) where xi=(i-3)/2 for i=1,...,5. Show that ||X||1=sum i=1 to n of the absolute value of Xi. Show that ||x||infinity= max (1<=i<=n) of the absolute value of Xi. Thank you for your explanation.