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# Linear Algebra

### Determinants and Cramer's Rule : Row Operations and Effect on Determinant, Computing Determinants and Solving Systems of Equations

1. State the elementary row operation being performed and its effect on the determinant Start with matrix a b c d a. c d a b b. a b kc kd c. a + kc b + kd c d 2. Compute the determinant a. 3 0 4 2 3 2 0 5 -1 b. 1 3 5 2 1 1 3 4 2 c. 3 5 -8 4 0 -2 3 -7 0 0 1 5 0 0 0 2 3. Use row reduction to convert t

### Systems of Linear Equations : Row Operations and Solutions

1. Find the augmented matrix for each system of linear equations: a. 5x1 + 7x2 + 8x3 = 3 -2x1 + 4x2 + 9x3 = 3 3x1 - 6x2 + x3 = 1 b. 4x1 + x2 - 7x3 = 6 5x1 + 7x2 + 2x3 = 3 5x1 + 2x2 + 5x3 = 7 c. 3x1 - 2x2 + 2x3 = 7 5x1 + 7x2 + 3x3 = 3 -5x1 + 6x2 - 8x3 = -5 2. Using elementary row operations reduce each of the augm

### Vector and Matrix Operations : Row Vectors, Column Vectors, Dot Product and Echelon Form

1. Find the dot product for the following pairs of vectors: a. Row vector = (2 0) Column vector is below 5 18 b. Row vector = (3 9 -4) Column vector is below 3 0 2 c. Row vector = (5 6 7 8) 1 1 1 1 The following matrices will be used in problems 2-3 below: 0 -2 5 A= 3 -4 17 1 2 3 9 7 2 -3

### Bijection between open interval and half-open interval

Please help with the following problem. Provide step by step calculations. Prove that there is a bijection from the open interval (0, 1) to the half-open interval (0, 1].

### Linear functional on N U 0

Let H = l^2(N U 0) (a) Show that if {a_n} is in H, then the power series sum_{n=0}^infty a_n z^n has radius of convergence >= 1. (b) If |b| < 1 and linear functional L: H-->F (F is either the real or the complex field) is defined by L({a_n}) = sum_{n=0}^infty a_n b^n, find the vector h_0 in H such that L(h) = < h, h_0 >

### Working With Matrices

Question: Any matrix B which is formed by the eigen vectors of a matrix A reduces the given matrix A to the diagonal form by the transformation (inverse of B)AB. i.e., (inverse of B)AB = diagonal matrix Please view the attachment to see the fully formatted problem.

### System of Equations and Linear Programming : Solving Subject to Constraints

1. Solve the following system of simultaneous equations: 6x1 + 4x2 = 40 2x1 + 3x2 = 20 x1 = ??? put your answer in the form x or x.x 2. Max Z = 5x1 + 3x2 Subject to: 6x1 + 2x2 <= 18 15x1 + 20x2 <= 60 x1 , x2 >= 0 Find the optimal profit. Z=? put your answer in the form x.xxx 3. Consider the

### 20 Systems of Equations Problems : Gauss-Jordan, Matrix Inverses, Production Matrices, Echelon Method and Application Word Porblems

(See attached file for full problem description) --- 1. What is true about the number of solutions to a system of m linear equations in n unknowns if m = n? If m < n? If m > n? Solve each system by the echelon method. 3. 2x + 3y = 10 -3x + y = 18 5. 2x - 3y + z = -5 x + 4y + 2z = 13 5x + 5y + 3z = 14 S

### To show that a given matrix is idempotent

Linear Algebra Matrices (X) Idempotent of a Matrix Show that the matrix A = [2, -1 , 1; -2 , 3 , -2; -4 , 4 , -3]

### To show a given matrix is orthogonal

Linear Algebra Matrices (IX) Orthoganal Matrix Show that the matrix A = [-8/9, 4/9 , 1/9; 1/9 , 4/9 , -8/9; 4/9

### Find the eigen values and eigen vectors of the matrix

Find the eigen values and eigen vectors of the matrix A = [9, -1 , 9; 3 , -1 , 3; -7 , 1 , -7] Also find the corresponding eigen spaces for the matrix A. Find a matrix B which reduce the given matrix A to the diagonal form by the transformation B^(-1)AB.

### Estimate the linear trend equation for the production series

Estimate line and find linear trend equation of the attached. A. Estimate the linear trend equation for the production series by drawing a line through the data. B. What is the average annual decrease in production? C. Based in the trend equation, what is the forecast for the year 2005? (See attached file for full pr

### Advanced Linear Algebra : Nilpotent, Vectors, Basis and Linear Independence

Let T: IR^3 IR^3 , T(x,y,z) = (o,x,2y) - Show that T is nilpotent of index 3 (that is, T^3 = 0 and T^2 different from 0 ) - Find a vector v E IR^3 s.t T^2(v) different from 0 and show that B = { v, T(v), T^2(v) } is linearly independent (so basis) - Find A = [ T ] an

### Computer Applications & Substitution, Elimination and Matrices

What computer applications can be used to graph systems of equations? Have you ever personally used any applications to graph an equation? In what circumstances does it make more sense to graph a system than to use the substitution, elimination, or matrix methods?

### Systems of Equations : Supply and Demand Equatons, Gaussian Elimination and Computer Applications

1) Solve the following system of equations that model supply and demand for a product: p - q = 0 ( supply equation ) cp - q = -1 ( demand equation ) where p = price and q = quantity Solve first when c = 0.999 and a second time when c = 1.001. What does the difference in solutions suggest about t

### Characteristic equation of a matrix and inverse of a matrix

Linear Algebra Matrices Eigen Values and Eigen Vectors(I) Find the characteristic equation of the matrix A = [1, 3, 7; 4, 2, 3; 1, 2, 1] Show that the equation is satisfied by and hence obtain the inverse of the given matrix. The fully formatted problem is in the attached f

### Pre Algebra Equations

Please see attachment. Thank you very much!

### Systems of Equations and Linear Programming: Six Word Problems

See the attached file. 1. A small school held a fund raiser. A total of 23 adults and children attended. Each adult paid \$10, and each child paid \$2 to attend. If a total of \$86 was raised, use a system of equations to determine how many adults and how many children attended. Solve the system using Gaussian Elimination. 2.

### Solutions of the system of equations

Determine the solutions of the system of equations whose matrix is row equivalent to 1 0 &#8722;1 |1 0 1 3 |1 0 0 0 |0 Give three examples of the solutions. Verify that your solutions satisfy the original system of equations.

### System of Equations Application Word Problem : Quarters and Dimes

Can I please find out what I am doing incorrectly with the following problem? "I have seventy five coins in my pocket made up of quarters and dimes. If the total value of the coins is \$12.45, how many of each coin do I have?" q + d = 75 Let q represent quarters (0.25)q + (0.10)d = 12

### Word Problems as Systems of Equations and Solving the Systems

Write a system of equations and then solve the following using matrix inverses. Consider a company that specializes in gourmet chocolate baked goods-chocolate muffins, cookies and brownies. Each chocolate muffin requires 2 units of chocolate, 3 units of flour and 2 units of sugar. Each chocolate cookie requires 1 unit of c

### Proof of the linear independence of exponential functions.

Let a_1, ..., a_n be distinct real numbers. Prove that the functions U_i(x) = exp (a_i x) are independent.

### Equation of an Ellipse, Interest and Lotto Probability

Several state and federal government spokespersons who use linear equations in a variety of ways. 1. An employee of the National Parks Service told you about a location in Washington, DC. It is a large grassy area south of the White House known as the Ellipse. The National Tree Lighting Ceremony is held annually on the Ellipse

### Linear Algebra: 7x7 Matrix, Adjoint and Conjugacy

Let A be a 7x7 matrix. Find all solutions of the equation A^7 = 0 up to conjugacy.

### Linear Algebra: Map on Space of all 5x5 Symmetric Matrices

Let L(A) = A + A^t be a map on the space of all 5x5 symmetric matrices. Prove that L is a linear operator and find its matrix in some basis.

### Slope and Intercept Calculations - Women, Pay and Education

Real Life Application of Linear Equations: A mathematician has developed a model indicating a woman with x years of education can expect to earn y = 1200x + 6300 dollars a year. What is the y-intercept for this model? Describe what it means in terms of the model. What is the slope for the model? Describe what this means in

### Application of Linear Equations Word Problem : Fixed Cost

A manufacturer finds that the cost y of producing x units is given by the formula of the form y=mx+b. If it costs \$1300 to produce 20 units and \$1750 to produce 35 units, what is the fixed cost?

### Method of Substitution vs. Method of Addition or Elimination for Systems of Equations

Please discuss the following: - Explain why vectors QR and RQ are not equivalent. - Explain when the elimination method for solving a system of equations is preferable to the substitution method. Give examples.

### Factorize

Factorize 513833.

### Matrix Representation of a Linear Operator on Vector Space

1) Let A be a linear operator on n-dimensional vector space V if {see attachment}. For some {see attachment}, prove that matrix representation {see attachment} of A with respect to a basis ... Please see attachment for complete question.