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# Linear Algebra

### Numerical Linear Algebra: Complex nxn Matrix

Let A = (aij ) be a complex n × n matrix. Assume that h Ax, x i = 0 for all x Є C n . Prove that (a) aii = 0 for 1 ≤ i ≤ n by substituting x = ei (b) aij = 0 for i 6 = j by substituting x = pei +qej then using (a) and putting p, q = ± 1, ± i (here i = √- 1) in various combinations Conclude that A = 0.

### State-Space Model

Obtain a state-space model of the system shown in Fig. 3 (see attached file).

### Linear Algebra Explanation

Hello all! I humbly apologise in advance if I'm posting inappropriately, or asking for a solution that is technically beneath you. Problem is as follows: "When Mary was half as old as Sam was when Mary was as old as Sam was when Mary was as half as old as Sam is now, Sam was four times as old as Mary was when Sam was

### Numerical Linear algebra. Gram-Schmidt orthogonalization.

See attached file for full problem description.

### Orthogonal Matrix and Eigenvalues : Let A be a 3 x 3 real orthogonal matrix with det A = 1. Prove that &#955;= 1 is an eigenvalue of A.

Let A be a 3 x 3 real orthogonal matrix with det A = 1. Prove that &#955;= 1 is an eigenvalue of A.

### College Algebra Functions and Graphs

1. To prepare, you must perform the following tasks: The following data was retrieved from www.cdc.gov. It represents the number of deaths in the United States due to heart Disease and cancer in each of the years; 1985, 1990, 1995, and 2002. Year Disease 1985 1990 1995 2002 Heart Disease 7711

### Linear equations

It is approximately 300 miles from Chicago, Illinois, to St. Louis, Missouri. Allowing for various traffic conditions, a driver can average approximately 60 miles per hour. How far will I need to travel to reach St. Louis after I have traveled 3 hours, and write a linear function that expresses the distance to be traveled to r

### Solve Systems of Equations : x + 3y = 12 and 4x - y = -17

The solution to the system of equations x + 3y = 12 4x - y = -17 is: A. (3,3) B. (12, -17) C. (-3,5) D. (5,3)

### Find the linear Equation relating book value and number of years.

An item costs \$1300, has a scrap value of \$100, and a useful life of six years. The linear Equation relating book value and number of years is: A. BV = -100x + 1300 B. BV = -100x + 1200 C. BV = -200x + 1200 D. BV = -200x + 1300

### Problem : An item costs \$900, has a scrap value of \$50, and a useful life of five years. The linear equation relating book value and number of years is:

An item costs \$900, has a scrap value of \$50, and a useful life of five years. The linear equation relating book value and number of years is: A. BV = -50x + 850 B. BV = -50x + 900 C. BV = -170x + 850 D. BV = -170x + 900

### Solve Simultaneous ( Systems of ) Equations

Solve for x and y in the following two sets of simultaneous equations: 4x-2y = 1 ......(i) 8x-4y = 1 ......(ii) y = 2x + 3.......(i) 2y - 4x = 6 .....(ii)

### Current Electricity: Kirchhoff's Laws (ully explained)

(See attached file for full problem description) --- 2. Finding Unknowns Determine the unknowns in the circuit shown in Fig.3. How do electrical engineers use Kirchoff's current law and Kirchoff's voltage law to write a system of linear equations? ? Based on your explanation, write a system of linear equations for Pro

### Least squares solution

Using the least squares method answer the following in a word doc. · What are the main strengths of this method? · What are its main shortcomings? · When would least squares be useful in the real world? · Does the use of linear algebra make this method easier to understand or use?

### Linear problem

I think the answer to this problem is true because 2x1 comes first and that coincides with 200. If I am wrong please explain. The question is The constraint 2x1 - x2 = 0 passes through the point (200,100). True or False.

### Numerical Linear Algebra : Unitary and Triangular Matrices and Krylov Matrix

1. Let A = QR be the factorization of a A into the product of a unitary matrix and a trianglular matrix. Suppose that the cloumns of A are linearly independent. Show that |rkk| is the distance from the k-th column of A to the linear space spanned by the first k-1 columns of A. 2. Let A &#1028; C^(mxm) and b &#1028; C^m be abi

### Eigenvalues : Asymptotic Stability of a System

Determine the asymptotic stability of the system x' = Ax where A is 3 x 3 matrix A = -1 1 1 0 0 1 0 0 -2 ( first row is -1 1 1 second is 0 0 1 and third is 0 0 -2) More specifically, what stability conclusion(s) can be drawn? ( Justify your answer)

### Repeated Eigenvalues System of Linear Differential Equations

Find the general solution for the following system: dx/dt = 3x - y dy/dt = 9x - 3y

### Exponential Random Variables with Parameters

It is known that the time (in hours) between consecutive traffic accidents can be described by the exponential r.v. X with parameter (Lambda = 1/60). Find (i) P(X < or = 60); (ii) P(X> 120); and (iii) P(10<X< or = 100).

### 25 Algebra Problems : Trigonometry, Graphing, Matrices, Simplifying Expressions and Solving Equations

The flow rate V (in cm/s) of a storm drainpipe follows the following expression. Find the value of V when t = 2.500 s. a. 33.56 b. 5.42 c. 132.6 d. 22.5 A boat approached a harbor and the captain takes two bearings to calculate the distance to the harbor. Bearing 1 is read at 0 degrees and bearing 2 is read at 30 de

### Three Variable Systems of Equations from Application Problems

1. The sum of three numbers is 6. The third number is the sum of the first and second numbers. The first number is one more than the thrid number. Find the numbers. 2. Sports - Alexandria High School scored 37 points in a football game. Six points are awarded for each touchdown. After each touchdown, the team can earn

### Discontinuous Counter Example

I need a counterexample for the following: If f:[a,b] -> R is ONE-TO-ONE and satisfies the intermediate value property, then f is continuous on [a,b]. I know that this is a false statement if you exclude the one-to-one property. The example I received before was f(x) = sin(1/x), but this function is not one-to-one. I am

### Mathematical modeling using systems of linear first order equations

Two tanks each hold 3 liters of salt water and are connected by two pipes (see figure below) the salt water in each tank is kept well stirred. Pure water flows into tank A at a rate of 5 liters per minute and the salt mixture exits tank B at the same rate. Salt water flows from tank A to tank B at the rate of 9 liters per min

### Discuss each of the three operations in Gaussian elimination

1. Discuss each of the three operations in Gaussian elimination. What do they have in common and how do they help solve systems of linear equations? 2. What is an Eigenvalue? What is an Eigenvector?

### Linear Algebra : Leontief Input-Output Model and Real-World Applications

1. Why did Leontief use linear algebra techniques to create his model? Can you think of alternative methods? 2. What are the main strengths of his model? 3. Does it have any limitations (that you can think of)? 4. How might the Input-Output model be useful in the real world? (In other words, would anyone except an Economist

### Mathematical System

I am having a problem drawing the table for the following system: Define a universal set U as the set of counting numbers. Form a new set that contains all possible subsets of U. This new set of subsets together with the operation of set intersection forms a mathematical system. Then I have to tell which properties that we did

### Subsets in n-space

I've been having trouble with this and need some assistance. v - union ^ - intersection If S and T are subsets of Rn, prove that (int S) ^ (int T) = int (S^T) and (int S) v (int T) subset int(SvT)

### Gap between two consecutive squarefree numbers can be arbitrary

Problem 5. Recall that a number n is called squarefree if it is not divisible by any square > 1. Show that the gap between two consecutive squarefree numbers can be arbitrary large. (Hint: Find a positive integer m such that m is divisible by 2^2, m + 1 is divisible by 3^2, m + 2 is divisible by 5^2, m + 3 is divisible by 7^2 an

### Systems of Equations Application and Systems of Inequalities

# 41 Nickels and dimes. Windborne has 35 coins consisting of dimes and nickels. If the value of his coins is # 3.30, then how many of each type does he have? # 43 Blending fudges. The chocolate factory in Vancouver blends its double-dark-chocolate fudge, which is 35% fat, with its peanut butter fudge, which is 25% fat, to

### Functions, relations, and linear equations

1. In the real world, what might be a situation where it is preferable for the data to form a relation but not a function? There is a formula that converts temperature in degrees Celsius to temperature in degrees Fahrenheit. You are given the following data points: Fahrenheit Celsius Freezing point of water 32 0

### Vector Spaces, Linear Transformations, and Eigenvalues / Eigenvectors, Basis and Diagonalizing Matrices

1. Answer the following questions A. Is lambda = 2 an eigenvalue of the following matrix? Why or why not? 3 2 3 8 B. Is lambda - -2 an eigenvalue of the following matrix? Why or why not? 7 3 3 -1 C. Is the following vector an eigenvector of the following matrix? If so, find the eigenvalue. 1 4 -3 1 -3 8 2. Find