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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.)

Diagonizable 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

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


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

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)

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


For which value of k does the matrix A = 1 k 4 8 have one real eigenvalue of multiplicity 2? K=___?

Find Associated Eigenvectors

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


Matrix A (-4k) has two distinct real eigenvalues if and only if k > ____? (19 )


? 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)


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.

Hi, I am having difficulties solving this problem. 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. Thanks.

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

Linear Differential Equations

Find the fixed points and sketch trajectories in the phase plane for the system: ... using the phase portrait, examine the behaviour of solutions of this system as t&#8594;&#8734; when they start from (x,y)=(-1,0), and when they start from (x,y)=(-1,-1). Please see attached for full question.

Convex hull

If A={(-1,-1),(3,-4),(-2,5),(0,3),(2,1),(4,7)}what would be the convexhull(A) expressed as the intersection of a minimum number of closed halfplanes. ALSO, if K is the intersection of the halfspaces: {(x,y,z):x>=0} {(x,y,z):y>=0} {(x,y,z):z>=0} {(x,y,z):x+2y+3z<=6} {(x,y,z):x+3y+2z<=6} {(x,y,z):x<=4} what would the ver

Isomorphisms and eigenvalues

True or false? Justify your answer in each case (giving a proof or a counterexample): Let T:V-->W be a linear transformation which is an isomorphism. Denote its inverse by T-1 . Suppose that (SYMBOL) is an eigenvalue of T. Then answer a, b, c and d. PLEASE SEE ATTACHMENT FOR COMPLETE QUESTION

Linear Transformation and Matrix

Please help me solve the following linear algebra questions involving linear transformation and matrices. (see attached) ? Let and let . Define a map by sending a vector to . a) let and be the standard basis vectors of V. let , and be the standard basis vectors of W. Find the matrix of T with respect to


? True or false? If U is a subspace of V then V-U = also is a subspace. (Proof or counterexample)

Non Linear System - Second Order System

1// Is it safe to state that for non linear system it's safe to assume/start off with x1 dot = x2? 2// The solutions to x1 dot and x2 dot are provided in the attached. I'm unsure how to arrive at said solutions and require assistance. Thanks in advance.