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# Linear Algebra

### Linear systems of quadratic equations and inequalities

Question One The relationship between the load on a reel, L kilonewtons per metre (L kNm 1) and the reel diameter, x metres, is modeled by a graph consisting of two parabolic arcs, AR and BC, as shown. Arc AR is part of the parabola L =px2 + qx +r Points D(0. 1, 2.025), E(0.2, 2.9) and F(0.3, 3.425) lie on arc AR. Setup a syst

### A second order system satisfies the differential equation:

A second order system satisfies the attached differential equation: Calculate the natural complete response Xn(t) of the system, provided that: a0 = 13; a1 = 4; x(0) = 1; dx(o)/dt = 4. See attached file for full problem description.

### Linear system

Solve the equation. Determine whether it is inconsistent, dependent, or neither. 3x - 2y = 0 9x - 8y = 7

### Linear system

Solve the linear system. State whether the system is inconsistent, dependent, or neither. x/6 + y/3 = 8 x/4 + y/2 = 12

### System of nonlinear equations

If a system of nonlinear equations contains one equation whose graph is a circle and another equation whose graph is a line, can the system have exactly one solution? If so, what does the graph of the situation look like?

### Non-linear system

The following is a non-linear system. Solve it 1/x + 2/y = 3 2/x + 1/y = 4 (Hint: Try a change in variable. Let u = 1/x ; let v = 1/y)

### Linear Algebra with Cubic Roots

Exercise. IV. This problem is a partial investigation of which n×n matrices over C have cube roots; that is, for which n × n matrices A over C there is an n × n B over C such that A = B3. Since C is algebraically closed, every n × n matrix over C is similar over C to a matrix in Jordan canonical form. A. Suppose that A

### Complex Matrix, Diagonal Matrix, Left and Right Eigenvectors

5. Let X^-1 AX = D, where D is a diagonal matrix. (a) Show that the columns of X are right eigenvectors and the conjugate rows of X^-1 are left eigenvectors of A. (b) Let ... be the eigenvalues of A. Show that there are right eigenvectors x1,. . . , x and left eigenvectors y1, . . , yn such that A =... keywords: matrices

### Eigenvalues, eigenfunctions and modified Green's function.

5.6. (a) Find the eigenvalues and eigenfunctions of ?u"=&#955;u, ?1<x<1; u'(1)?u(1) =0, u'(?1)+u(?1) =0 Show that there is precisely one negative eigenvalue, that zero is an eigenvalue, and that there are infinitely many positive eigenvalues. Show graphically how the eigenvalues are determined. (b) Find the modified Green's f

### Differential Equations

(See attached file for full problem description) 1) The slope field for the system dx/dt = 2x + 6y dy/dt = 2x - 2y is shown to the right a) determine the type of the equilibrium point at the origin. b) calculate all straight-line solution. 2) show that a matrix of the form A =(a b; -b a) with b!=0 must have complex eig

### eigenvalues and eigenvectors of matrix

I have some very basic lin algerbra eigenvalue problems. (See attached file for full problem description) 1. Find the eigenvalues and eigenvectors for the projection matrix P = [0.2 0.4 0; 0.4 0.8 0; 0 0 1]; 2. Find the eigenvalues for the permutation matrix P = [0 1 0; 0 0 1; 1 0 0]; 3. Finish the last row to make the mat

### Symplectic Matrix, Eigenvalues and Multiplicity

A 2n x 2n M is symplectic if where J is the (also 2n x 2n) matrix . Prove that if is an eigenvalue of M , then so is , and that these have the same multiplicity. Show furthermore that, if are eigenvalues of M, and , then the corresponding eigenvectors have the property that Please see the attached file for th

### Pairwise Sequential Voting

A seventeen-member committee must elect one of four candidates: R, S, T, or W. Their preference schedule as shown below. Which candidate wins under pairwise sequential voting with the predetermined order S, T, W, R? Number of Members Ranking 6 R > S > T > W 5 S > R > T > W 3

### Banach Space and Closed Subspace

Let I = [a,b] be a finite interval. Show that the space C(I,R^n) of continuous functions from I into R^n is a Banach space with the uniform norm llull = sup l u(t) l where t is in I. (Show that this is a norm and that C(I,R^n) is complete). See attached file. Please be very detailed when answering question.

### Modules, Linear Operators, Characteristic & Minimal Polynomials

See the attachments. Let F be a field and . Then is an n - dimensional vector space over F. Define a function by . (a) Show that T is a linear operator. (b) Find the characteristic and minimal polynomials for T, with explanation. (For the characteristic polynomial, recall that you will need to choose a basis for

### Linear Programming Proof

I need the proof of the Linear programming problem attached. --- Consider the LP: Min ct x Subject to Ax &#8805; b, x &#8805; 0. One can convert the problem to an equivalent one with equality constraints by using slack variables. Suppose that the optimal basis for the equality constrained problem is B. Prove t

### Properties of Condition Numbers : Orthogonal Matrices and Eigenvalues

Please prove the properties of condition numbers attached to this message. Refer to definitions/theorems you used. Also, if you want, have a look at the second file attached, since I believe that you can refer to the previous properties to do 6 to 10. 7. For any orthogonal matrix Q, i2(QA) = k2(AQ) = k2(A) 8. If D= diag(d1,

### Linear Algebra : Use Network Analysis to Determine Number of Traffic Sensors

A traffic engineer wants to know whether measurements of traffic flow entering and leaving a road network are sufficient to predict the traffic flow on each street in the network. Consider the network of one-way streets shown in the Figure 3. The numbers in the figure give the measured traffic flows in vehicles per hour. Assume

### Linear Algebra : Solving for Temperatures of Points on a Flat Square Plate

The concept of thermal resistance described in Problem 5 can be used to find the temperature distribution in the flat square plate shown in Figure 5(a). Figure 5(a) The plate's edges are insulated so that no heat can escape, except at two points where the edge temperature is heated to Ta and Tb, respectively. The temperat

### Linear Algebra : Calculating heat loss through a wall

Engineers use the concept of thermal resistance R to predict the rate of heat loss through a building wall in order to determine the heating system's requirements. This concept relates the heat flow rate q through a material to the temperature difference &#8710;T across the material: q = . This relation is like the voltage-curr

### Bounded Linear Operator: Bounded Invertible an Norm

11.8 Let and where a "1" appears in the n-th position and a zero in all other positions. Let (an) be a sequence of complex numbers. Prove then that (i) ... defines a bounded linear operator on G if and only if... , and accordingly find the norm of T. (ii) What are the necessary and sufficient conditions for T to be

### Linear and Non-Linear Equations : Finding Minimum or Maximum Value

1) An open-top box is to be constructed from a 4 by 6 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out. a) Find the function V that represents the volume of the box in terms of x. Answer: b) Graph this funct

### Systems of Equations and Inequalities Applications Word Problems

1. Solve the system of equations by elimination. 7x + 8y = -55 4x + 5y = -34 2. Ron and Kathy are telemarketers. Ron contacts potential home buyers and is paid \$30.00 for each buyer he gets to work with a realtor at the company. Kathy contacts potential sellers and is paid \$65.00 for each seller she gets to discuss l

### Linear Algebra - Linear Functionals

See the attached file. Now let V be the space of all 2x2 matrices over the field F and let P be a fixed 2x2 matrix. Let T be the linear operator on V defined by T(A) =PA. Prove that tr(T)=2tr(P). From a previous exercise we know that similar matrices have the same trace. Thus we can define the trace of a linear operator o

### Linear Algebra: Dual Space and Dual Basis

Let β = {α1, α2, α3} be the basis for (complex) (C^3) defined by α1=(1,0-1), α2=(1,1,1), α3=(2,2,0). Find the dual basis of β showing all work. Please see the attached file for the fully formatted problems.

### Non-linear Differential Equation Word Problem

Two chemicals A and B are combined to form a chemical C. The rate of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to chemical C. Initially there are 40 grams of A and 50 grams of B, and for each gram of B, 2 grams of A are used. It is observed that 10 grams of C are formed

### Basis and basis matrices

3. Let B1 = {v1,v2,v3} be a basis of vector space V and B2 = {w1,w2,w3} where w1=v2+v3; w2=v1+v3; w3= v1+v2 Verify that B2 is also a basis of V and find the change of basis matrices from B1 to B2 and from B2 to B1. Express the vector a(v1) +b(v2) + c(v3) as a linear combination of w1,w2,w3

### Linear algebra

Please help with the following problems. 1. Let u1 = (1,2,1,-1) and u2 = (2,4,2,0). Extend the linearly independent set {u1,u2} to obtain a basis for R4 (reals in 4 dimensions) 2. Let U1,U2 be two subspaces of a finite dimensional vector space V such that U1+U2 = V. Prove that there is a subspace W of U1 such that W (+)

### Systems of Three Equations with 3 Variables ( 3 Equations and 3 Unknowns )

Please help me solve the following system of three equations and discribe the methods that are being used to help me understand: X + Y + Z = 6 2X - Y + 3Z = 8 3X - 2Y - Z = -17 Thank you!

### 1. Solve this system of four equations: (see full description) 2. If x varies directly as 2y-1, and x=9 when y=2, find the value of y when x has a value of 15

1. Solve this system of equations. Put your answer in alphabetical order. 5w+3x-y-z=-2 w+x+y+z=2 2W-x+y+4z=0 3w+2x+3y+2z=1 2. If x varies directly as 2y-1, and x=9 when y=2, find the value of y when x has a value of 15.