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Linear Algebra

Linear equations and inequalities explained in this solution

Part 1: What is the formula for the volume of a rectangular solid? Find an object in your residence that has the shape of a rectangular solid. Measure and record the length, width, and height of your object in either centimeters (to the nearest 10th of a centimeter) or inches (to the nearest quarter of an inch). Compute the vol

Linear Algebra : Orthogonal Basis

Please see the attached file for the fully formatted problems. Problem a. quote a theorem which guarantees that there exists an orthogonal basis for (with standard inner product) made up of eigenvectors of matrix b. Find such a basis . c. Represent the quadratic form by a symmetric matrix. Is Q positive definite?

Linear Algebra : Linear transformations, Diagaonalization and Adjoints

Let V be a , but with the weighted inner product, , ,where and . Let be the linear transformation given by T(a,b,c)=(3a-2c,b,3a+10c). a. Show that T can be diagonalized and find a basis for V comprised of eigenvectors of T. b. Find the matrix of the adjoint of T with respect to the basis . Please see the atta

Linear Algebra : Symmetric Polynomials and Inner products

Let be the real vector space of "symmetric" polynomials of degree at most 4, with inner product a. find a basis for V and determine dim V. b. viewing V as a subspace of R) with the same inner product, find the "closest" point in V to the polynomial . Please see the attached file for the fully formatted problems.

Complementary Angles and Systems of Equations

Two angles are complementary of each other. Twice one angle is equal to the other angle plus the product of three and five. 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: Y

Linear Operators : Finite-dimensional Vector Space, Fields and Mappings

Let V be a finite-dimensional vector space. The base field F may be either R or C here. Let T, an element of the linear mapping of V to V, L(V), be an operator. Suppose that all non-zero elements of V are eigenvectors for T. Show that T is a scalar multiple of the identity map, i.e., that there is a λ in the Reals such

Evaluating: Dimension and Null Space

If you were to let A be a 6 x 14 matrix where the dimension of the row space is 3 (dim(R(A) = 3), what would the dimension of the null space of matrix A (dim(N(A)) be and what would the dimension of the null space of A^T (dim(N(A^T)) be? Make sure to show all work involved.

Linear Algebra : Subspace

If U how would you show U is a subspace? Also, how would you find a subspace V of such that U, V such that X = U + V? Please see the attached file for the fully formatted problems.

Linear Algebra : Change of Coordinates

If you let B = {v1, v2, ..., vk} be a basis of a subspace V of ; and you let Q = (qij) be an n x n matrix such that C = {Q(v1), Q(v2,)...,Q(vk)} is a basis of V. If , what are the coordinates of v with respect to B? Also, if what are the coordinates of Q(v) with respect to C?

Linear Algebra: Null Space and Column Space

I have attached a word document that contains my question. In the attached document R( ) is the row space, N( ) is the null space, and C( ) is the column space. If you were to let A be a 6 x 14 matrix where the dimension of the row space is 3 (dim(R(A) = 3), what would the dimension of the null space of matrix A (dim(N(A)) be

Systems of Equations Word Problems

Supppose a baseball is thrown at 85 miles per hour.The ball will travel 320 ft when hit by a bat swung at 50 miles per hour and will travel 440 ft when hit by a bat swung at 80 miles per hour. Let y be the number of ft traveled by the ball when hit by a bat swung at x miles per hour.(Note: The precceding data is valid for 50 les

Linear Algebra : Linear Combinations

If you assume {v1, v2, ..., vk} and , and you also assume {v1, v2, ..., vk} are linearly independent and {v1, v2, ..., vk, w} are linearly dependent. How would you show that w can be uniquely expressed as a linear combination of {v1, v2, ..., vk}? Also, if the zero vector is included among the vectors {v1, v2, ..., vk}, w

Systems of Linear Equations

Solve the following model for the prices of two goods, tea and coffee, demonstrating that one gets the same answer by using either variable elimination or matrix algebra. You must use both methods. Show all steps. The price of tea is Pt and the price of coffee is Pc. Quantities are assumed to adjust outside the model. Pt=8Pc

Principal Ideal Domains

Please see attached file. The file got cut off a bit, but should read "For any prime 'p' of R prove that....".

Systems of Equations Substitution and Addition Methods

Solve the following system by graphing: 14. x - 2y = -6 y = -3x/2 - 1 15. y < 5x - 2 y > 3x - 2 Solve the following systems by the addition method: 16. x + 2y = 4 3x - 6y = 6 17. 4x - 5y = 20 y = 4/5x - 4 Solve the following systems by the substitution method: 18. 7x - 4y = 26 y =

Heating Water : Modeling with a Linear Equation

Water is the most importnt substance on Earth. One reason for its usefulness is that is exists as a liquid over a wide range of temp. In its liquid range, water absorbs or releases heat directly in proportion to its change in temp. Consider the following data that shows temp of a 1,000 g sampe of water at normal atmospheric pres

Linear Algebra : Complex Vector Space and Eigenvalues

This is problem #15 on page 189 of Axler's book Linear Algebra Done Right. Suppose V is a complex vector space. Suppose T is in L(V) is such that 5 and 6 are eigenvalues of T and that T has no other eigenvalues. Prove that (T &#8722; 5I)^(n&#8722;1)*(T &#8722; 6I)^(n&#8722;1) = 0, where n = dimV.

Systems of Linear Equations

Please go over one more time. Find the values of x and y that solve the following systems of equations. -8x - 9y = -25 -4x + 3y = -5 I think I am getting confused on the substituting.

Solve the equation by graphing.

These are questions that I got wrong on a test..can someone show me how to do these? ________ Solve the equation by graphing. 2x-3y=12 3y-2x=-12 is this equation independent inconsistent or dependent? __________________________________________ Solve the equation by graphing. 3x-y=4 3x-y=0 is this equation

Systems of Linear Equations

Please assist to understand the difference in these different methods. Please show work so I can follow the solution. Determine if the given ordered pair is a solution to the system: 12. 2x + y = 5 (4, -3) x - y = 1 22. 4x - y = -2 (-1, -2) 3x + y = -5 Solve each system of equations by the graphing method: