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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

For what value of the parameter b will the following system of equations fail to have a unique solution? (HINT - Do not attempt to actually solve the equations!!!!) x+2by-z = 2 2bx+3y-bz = 3 x+2y+z = 0

Linear Algebra

Compute the determinants of the following matrices? 0 2 1 -1 4 3 -2 1 -4 3 0 1 -4 1 -1 2 3 0 4 2 0 -2 1 0 1

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

Is K={f_1(x)=1, f_2(x)=sin x, f_3(x)=cos x }cC[0,1] linearly independent or linearly dependent? Justify your answer.

Euclidean Linear Dependence

1. Is G={ [1 -1], [1 -4], [1 -6], [0 0] cM^2(R) linearly independent or linearly dependent? Justify your answer. { [-1 0] [1 0] [1 0 ] [1 0]}

Normal Subgroups

Please see the attached file for the fully formatted problem. Let G be a group and let D ={(a,a,a):a E G}. Prove that D is a normal subgroup of G+G+G if and only if G is Abelian.

Vectors

Suppose that a "skew" product of vectors in R2 is defined by (u,v)=u1v1-u2v2 Prove that (u,v)squared >equal too (u,u)(v,v). (NOTE; This is just the reverse of the Cauchy- Schwartz inequality for the ordinary dot product.)