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Linear Algebra

Linear equations

1. Solve the inequality. Write the solution in interval notation and graph the set on the number line. -2(x - 4) 3x + 1 - 5x 2. Solve the following problem by writing an equation and then solving the equation: You invest $7,200 in two accounts paying 8% and 10% annual interest respectively. At the end of the year, the acco

Linear Algebra : Invertible Matrices

Let A be the real 2x2 matrix [a b] [c d] with bc greater than or equal to 0. Prove there exists a real 2x2 invertible matrix S so that S^-1 A S is either diagonal or of the form [x 1] [0 x] where x is the eigenvalue of A.

Linear Algebra : Diagonalizing Matrices

Let B be an nxn matrix with B^2 = B prove that B is diagonalizable, ie there exists an invertible matrix S so that S^-1 B S is diagonal. (Hint: all eigenvalues of B are either 0 or 1. For each k between 0 and n, consider the case when the nullity of B is k.)

Linear Algebra : Hermitian Similar Matrices

Suppose A & B are Hermitian matrices and AB=BA, show that A and B are simultaneously diagonalizable, ie, there exists an unitary matrix C so that both C*AC adn C*BC are diagonal.

Diagonalizable Matrix, Inverse and Nullspace

1) If a Matrix A is diagonizable, must it have an inverse ? if so, is it diagonizable? Can {see attachment} be diagonized, does it have an inverse as well as {see attachment} 2) A is mxn For m<n, is there a vector b such that Ax = b does not have any solution? Any trivial solution for Ax = 0? b) Can say the same for m>n ? A

Eigenfunction Problem

Given y " + ky = 0; y(0)=0 and y'(1)=0; a) Determine the normalized eigenfunction for this problem; b) Use the results in part (a) to express f(x)=x in an eigenfunction expansion, i.e. determine the expansion coefficients (Cn).

Eigenvalues

Calculate the eigenvalues of this matrix: -16 6 60 2 [Note-- you'll probably want to use a graphing calculator to estimate the roots of the polynomial which defines the eigenvalues..... (see attached)]

Eigenvalues of Matrix

Calculate the eigenvalues of this matrix: {see attachment} Note: You'll probably want to use a graphing calculator to estimate the roots of the polynomial which defines the eigenvalues.

General Real-Valued Solutions

Find the general real-valued solution of each system. Classify the origin as a saddle, center, spiral, or one of the normal types (identify the type). Identify as neutrally stable, unstable, or asymptotically stable. *See attachment for systems

Field Condition Sets

LET F be a field and set G = a b -b a : a,b is an element of F. Under what conditions on F will G be a field? Can you give an example of such F other than R (real numbers)?

Differential Equations for Lead-Free Subjects

Suppose that the initially lead-free subject in Example 6.1.1 is exposed to lead for 400 days, and then removed to a lead free environment. Use a computer to estimate how long it takes the amount x3 of lead in the bones to decline 50% of x3(400); repeat for 25% and 10%. (See attachment for full details)

Eigenvectors : Linear Independence

The matrix A = 1 1 0 0 0 0 0 1 1 has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and the eigenvectors. Eigenvalue of multiplicity 1 : 0 Associated Eigenvector: .57735 -.57735 .57735

Vectors : Linear Independence

Let A = [ -5, 6, -19] B= [-1, 2, -3] and C= [-2, 2, -8] **they are linearly independent*** The problem is the same as if the vectors were written vertically. If they are linearly dependent, determine a non-trivial linear relation - (a non-trivial relation is three numbers which are not all three zero.) otherwise, if t

Eigenvalues and Eigenvectors

The matrix A = 1 1 0 0 0 0 0 1 1 has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and the eigenvectors. Eigenvalue of multiplicity 1 : Associated Eigenvector: Eigenvalue of multiplicity 2 : Associated two linearly independent

Find Associated Eigenvectors

A = -14 -4 20 4 smaller eigenvalue = -6 associated eigenvector= (__ , __) larger eigenvalue = -4 associated eigenvector= (__ , __) Find associated eigenvectors.

Proof of diagonalizability

Verify: (a) If A is diagonalizable and B is similar to A then B is also diagonalizable. (b) If {see attachment} and x is an eigenvector of A corresponding to an eigenvalue ... {see attachment for complete question

Homomorphisms

? Let G be a group and let a,b be two elements of G. The conjugate of b by a is, by definition, the element . The centralizer of a, denoted by s the set of all elements g in G such that ga=ag. i) Find all possible conjugates f the permutation ii) Find the centralizer p in . iii) Prove that for any element a in a g

Solbing System of Linear Equations

1. Solve by substitution or elimination method: 3x - 2y = 8 -12x + 8y = 32 2. Solve by substitution or elimination method: 7x - 5y = 14 -4x + y = 27 3. Solve by substitution or elimination method: -4x + 3y = 5 12x - 9y = -15 4. A university boo

Solving Systems of Linear Equations

1. Why do intersecting lines represent a unique solution? Give examples to support your answer. 2. What is the significance of the name 'linear equation' to its graphical representation? 3. The solutions of line m are (3, 9), (5, 13), (15, 33), (34, 71), (678, 1359), and (1234, 2471). The solutions of line n are (3, -9)

Equations

1.Can you show that, given two equations y = m1x + c1 and y = m2x + c2 where c1 and c2 are different, there is no solution if m1 = m2. Interpret this result graphically. Also show that if c1 = c2 then there will be at least one solution no matter what m1 and m2 are. Interpret this result on a graph. 2.In your reading you have

Eigenvalues for Linear Algebra Class

Please show all the steps involved. 1. An nxn matrix A is said to be nilpotent if A^k = O (the zero matrix) for some positive integer k. Show that all the eigen values of a nilpotent matrix are O.

Matrix : Convergence, Pseudoinverse and Single Value Decomposition

Only problems #3 &4-a,(without using any software). 3 . Ax = b we consider the iterative scheme .... where the matrix Q is nonsingular. (a) If ... for some subordinate matrix norm, show that the sequence produced by the above scheme converges to the solution of the system for any initial vector x(0). 4. Given singular