Explore BrainMass

Linear Algebra

Fermat Numbers

The Fermat numbers are numbers of the form 2 ^2n + 1 = &#934;n . Prove that if n < m , then Φn │ϕ m - 2. The Fermat numbers are numbers of the form 2 ^2n + 1 = (Phi)n . Prove that if n < m , then (Phi)n │(Phi)m - 2.

Perturbed Linear System

Consider the perturbed linear system x' = (A + eB(t))x, x is an element of R^n, where A is a constant matrix, B is a bounded continuous matrix valued function, and e is a small parameter. Assume that all eigenvalues of A have non-zero real part. 1) Show that the only bounded solution of the system is 0. 2) If A ha

Systems of Equations : Five Word Problems

There are hens + rabbits. The heads = 50 the feet = 134. How many hens & how many rabbits ? Flies + spiders sum 42 heads and 276 feet. How many of each class? J received $1000 and bought 9 packs of whole milk & skim milk that totalled $960 - How many packs bought of each kind? A number is composed of two integers and its sum

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

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

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

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).

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 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.

Linear Algebra : Vectors - Inner Products

Given a vector w, the inner product of R^n is defined by: <x,y>=Summation from i=1 to n (xi,yi,wi) [a] Using this equation with weight vector w=(1/4,1/2,1/4)^t to define an inner product for R^3 and let x=(1,1,1)^T and y=(-5,1,3)^T Show that x and y are orthogonal with respect to this inner product. Compute the values of

Linear Algebra and Numerical Analysis Polynomials

Questions on a Sequence of Polynomials. See attached file for full problem description. Let be the sequence of polynomials defined by , , 1) Show that is a polynomial of degree k. Calculate the coefficient of of . 2) Show by induction that for all real . 3) Deduce that if , . 4) Show that for all whole nat

Linear Algebra: Linear Mapping

Consider the following linear mapping from C[-pi,pi] into itself: L(f)=integral from -pi to pi of G(x),h(y),f(y)dy for any function f(x) in C[-pi,pi]. Here G(x), H(x) are given continuous functions. Find a function f such that L*f=lambda*f for some lambda and find the value of lambda. This is a generalization of the notion

Linear Algebra: Vector Spaces

Consider R2 with the following rules of multiplications and additions: For each x=(x1,x2), y=(y1,y2): x+y=(x2+y2,x1+y1) and for any scalar alpha, alpha*x=(alpha*x1, alpha*x2) Is it a vector space, if not demonstrate which axioms fail to hold. Also, show that Pn- the space of polynomials of order less than n is a vector spac

Linear Algebra: Matrix of Transformation

Are the following examples linear transformations from p3 to p4? If yes, compute the matrix of transformation in the standard basis of P3 {1,x,x^2} and P4 {1,x,x^2,x^3}. (a) L(p(x))=x^3*p''(x)+x^2p'(x)-x*p(x) (b) L(p(x))=x^2*p''(x)+p(x)p''(x) (c) L(p(x))=x^3*p(1)+x*p(0)