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    Linear Algebra

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    Linear Algebra: Linear Transformations - Rotations

    Please see the attached file for the fully formatted problems. Calculate the standard matrix for each of the following linear transformations T. a. T: R^2 -> R^2 given by rotating -pi/4 counterclockwise about the origin and then reflecting across the line x_1 - x_2 = 0. b. T: R^3 -> R^3 given by rotating pi/2 counterclo

    Linear Algebra: Orthogonal Bases

    Let A be an n X n matrix and, as usual, let a_1, ..., a_n denote its column vectors. a. Suppose a_1,...,a_n form an orthonormal set. Show that A^-1 = A^T. b. Suppose a_1,...,a_n form an orthogonal set and each is nonzero. Find the appropriate formula for A^-1.

    Linear Algebra - Basis and Dimension

    Please see the attached file. Please show each step of your solution. Thank you. Find a basis for each of the given subspaces and determine its dimension. V = Span ((1, 2, 3), (3, 4, 7), (5, -2, 3)) C R^3

    Linear Algebra - Orthogonal Bases

    Please see the attached file for the fully formatted problems. Let V = Span ((1, -1, 0, 2), (1, 0, 1, 1)) C R^4, and let b = (1, -3, 1, 1). a. Find an orthogonal basis for V. b. Use your answer to a to find p = proj_v b. c. Letting A = [ 1 1 -1 0 0 1 2 1] user your answer to b to give the least

    Linear Algebra - Four Fundamental Subspaces

    Please show each step of your solution. Thank you. Give a basis for the orthogonal complement of each of the following subspaces of R^4. a. V = Span ((1, 0, 3, 4), (0, 1, 2, -5)) b. W = {x E R^4 : x_1 + 3x_3 + 4x_4 = 0, x_2 + 2x_3 - 5x_4 = 0}

    Linear Algebra : Span and Subspace

    (The material is from Basis and Dimension. Please explain each step of your solution for # 8, 9. Thank you.) Proposition 3.7: Let V in Rn be a k-dimensional subspace. Then any k vectors that span V must be linearly independent and any k linearly independent vectors in V must span V.

    Linear Algebra - Four Fundamental Subspaces

    (The material is from the Four Fundamental Subspaces. Please show each step of your solution. ) Theorem 4.7: Let A be an m x n matrix. Then 1. R (A) ┴ = N (A) 2. N (A) ┴ = R (A) 3. C (A) ┴ = N (A┬) 4. N (A┬) ┴ = C (A)

    Linear Algebra : Four Fundamental Subspaces

    Please see the attached file. (The material is from the Four Fundamental Subspaces. Please show each step of your solution.) Proposition 4.6 states: Let V in Rn be a subspace. Then (V â”´) â”´ = V.

    Five Questions - Sequences, Linear equations etc.

    1) Use inductive reasoning to determine the next three numbers in the pattern: 3,12,26,45.....? (I don't seem to find a pattern?) 2) Find the counterexample to show that the following statemnet is incorrect. "the sum of any odd numbers is divisible by 4. ( I tried several odd numbers and the answer is always divisible by 4)

    Seven Problems on Determinants, Cramer's Rule and Matrices

    Seven practice problems are in the attached file. 1. Use Cramer's rule to solve the system x + y -z = -1 2x - 2y + 3z = -12 X + y - 4z = 13 Solution set is 2. Evaluate the determinate 5 0 0 1 -3 1 1 2 4 3. Use Cramer's rule to solve the system or to determine if it is inconsistent(I) or dependant(

    Converting temperature and modeling cost

    Find a record-breaking temperature (in degrees Celsius) for a town or city in a country other than the United States. Include the name of the town and country along with the temperature, and what record was broken. Give the formula for converting degrees Celsius to degrees Fahrenheit. Using the formula, show how to convert the m

    Manipulating Basic Linear Equations

    The following is offered as a solution of the equation -4[x-2(2x-3)]+1= 1/2(4x-6) -4[x-2(2x-3)]+1=8x-12 -4x-4x+6+1=8x-12 -8x+7=8x-12 7= -12 Because 7 = -12 is not a true equation, the equation has no solution. If this is correct, state that there is no solution. If not, explain in detail why it is not correct

    System of Linear Equations

    (1.) x-y+2z=13 2x+xy-z=-6 -x+3y + z =-7 a. Provide a coefficient matrix corresponding to the system of linear equations. b. What is the inverse of this matrix? c. What is the transpose of this matrix? d. Find the determinant for this matrix. (2) A = [2 -3] [-4 1] [7 4] and B=[ 6 5]

    Matrices, Inverse, Transpose, Determinant, Gaussian Elimination and Cramer's Rule

    Matrix methods can be used to solve linear programming problems. A linear programming problem is used to find an optimal solution, subject to stated restraints. 1. For example, consider an accountant who prepares tax returns. Suppose a form 1040EZ requires $12 in computer resources to process and 22 minutes of the accountant'

    Lines and equations

    Answer problems on attached word document. Answer Questions If possible show work. I have a lot more questions to answer and I need to use examples to work through them. 1. Find the slope for 3y - 1 = 14 2. Find the equation of the line in the form y = mx + b if possible if the line goes through (2, -3) and (-3, 4)

    Matrices and Row Reduced Echelon Form

    Please see the attached file for full problem description. 1. Five neighborhoods (NB) all want to raise money for a playground for their kids. The neighborhood that raises the most money will be able to choose the name of the park. To raise money, they all decide to have a bake sale and sell cookies (C), cakes (K), and muffin

    Prove that the fields R and C are not isomorphic.

    Check if the proof is correct. I need help to justify some of my answers by using Theorems, Definitions and etc. You can change my wording but try to stick to my idea. It's really important that you explain your work. Thanks! Note: R=Real Number and C=Complex Number

    Angles, Systems of Equations and Gaussian Elimination

    Two angles are supplementary of each other. Twice one angle is equal to the other angle minus the product of six and eight. A. Set up a system of linear equations to represent the two angles. (Hint: You will need two equations and two unknowns.) B. Graph each of the equations on one rectangular coordinate system. (Hint: