Let x (t),..., xk (t) be solutions of a homogeneous n-dimensional linear system x' = A(t)x,...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Let x₁ (t),..., xk (t) be solutions of a homogeneous n-dimensional linear system x' = A(t)x, where the matrix map t → A(t) = Rn×n is continuous. Show: if the vectors x1(0), ..., xk (0) are linearly independent vectors in R", then x₁(t),...,xk (t) remain linearly independent for all times. [Hint: consider suitable linear combinations and exploit uniqueness of solutions to IVPs.] Let x₁ (t),..., xk (t) be solutions of a homogeneous n-dimensional linear system x' = A(t)x, where the matrix map t → A(t) = Rn×n is continuous. Show: if the vectors x1(0), ..., xk (0) are linearly independent vectors in R", then x₁(t),...,xk (t) remain linearly independent for all times. [Hint: consider suitable linear combinations and exploit uniqueness of solutions to IVPs.]
Expert Answer:
Answer rating: 100% (QA)
Solution 2 V linear system So that Let we given tha... View the full answer
Related Book For
Differential Equations and Linear Algebra
ISBN: 978-0131860612
2nd edition
Authors: Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West
Posted Date:
Students also viewed these mathematics questions
-
Is it possible that all nontrivial solutions of a homogeneous system of 5 equations in 7 unknowns be multiples of each other? Explain.
-
Let x1,..., xk be linearly independent vectors in Rn, and let A be a nonsingular n n matrix. Define yi = Axi for i = 1,....,k. Show that y1,....,yk are linearly independent.
-
Let x1, x2,..., xk be linearly independent vectors in a vector space V. (a) If we add a vector xk+1 to the collection, will we still have a linearly independent collection of vectors? Explain. (b) If...
-
Describe the typical terrorist cell.
-
1. The film sequence shows three people interacting in a work environment. Which aspects of organizational behavior discussed in this chapter appear in this sequence? 2. The three people in this...
-
After the issuance of its audit report, a CPAfirm may encounter evidence indicating that the client's financial statements were materially misstated or lacked required disclosures. 1) What actions...
-
Consider the simulation flowsheets in Figure 7.36, which were prepared for ASPEN PLUS. The feed stream, S1, is specified, as are the parameters for each process unit. Complete the simulation...
-
To encourage its foreign managers to incorporate expected exchange rate changes into their operating decisions, Vancouver Enterprises requires that all foreign currency budgets be set in Canadian...
-
P = 7 kN, w = 12 KN/m and angle is 32 deg Equilibrium and support reactions for beams and frames Question 2a [25 marks] Show that the following beam is statically determinate. Determine the external...
-
Suppose that a customer were to purchase from Colonial Originals a furniture kit that has a catalog price of $99.95 and a shipping weight of 26.5 pounds. a. Using Table 3-3, determine the total cost...
-
Federal state and local income taxes have been deductible since the inception of the federal income tax in 1913. Proponents of the elimination of the state and local income tax deduction argue that...
-
On January 1 , 2 0 2 3 , Holland Corporation paid $ 7 per share to a group of Zeeland Corporation shareholders to acquire 6 0 , 0 0 0 shares of Zeeland s outstanding voting stock, representing a 6 0...
-
Briefly in two sentences each How does the War in Ukraine impact Asia/Austraila How does the economic dimension (look at strengths and weaknesses) get impacted in Asia How do you see the SC...
-
In the month of March, a department had 8 3 0 0 units in beginning Work in Process Inventory that were 6 0 % complete. During March, 3 0 6 0 0 units were transferred into production from another...
-
Write a program that accepts an integer limit and an integer value and that prints a complete line of output reporting the multiples of value uptil the limit . Keep prompting the user for a value...
-
Kubin Company s relevant range of production is 2 4 , 0 0 0 to 3 1 , 0 0 0 units. When it produces and sells 2 7 , 5 0 0 units, its average costs per unit are as follows: Average Cost per Unit Direct...
-
Test whether the following matrices are nonsingular: 1 (0) [ 7 -1 1 [ (a) 19 1-3 7 1 0 1 40 4-2 1 (b) -5 60 7 03 (c) (d) 13-3 -495 301 1086 0 4
-
If you want to solve a minimization problem by applying the geometric method to the dual problem, how many variables and problem constraints must be in the original problem?
-
Tandem with a Twist (a) Use the method of the previous problem to solve the systems (b) Explain how the calculation of part (a) can be used to solve the matrix equation For the unknown 3 ( 2 matrix...
-
Hard Puzzle the square root of a matrix A is a matrix R such that RR = A. Show that the matrix Has no square root, while the matrix Has an infinite number of square roots. 0 0
-
For what values of the nonnegative constant a in the equation y" + y = 0 do thereexist nonzero solutions satisfying the boundary condi-tions in Problems a-b? The values of a are called eigenvaltions...
-
Consider a binary equimolar mixture of acetone and chloroform at \(300 \mathrm{~K}\) and \(1 \mathrm{~atm}\). Because this mixture forms a binary azeotrope, methyl- \(n\)-pentyl ether solvent is used...
-
A system of three parallel reactions (Trambouze and Piret, 1959) involves the following reaction scheme: \[\mathrm{A} \xrightarrow{k_{1}} \mathrm{~B} \quad \mathrm{~A} \xrightarrow{k_{2}} \mathrm{C}...
-
Repeat Exercise 8.1, taking the first two reactions as first order, and the last as second order with \(k_{1}=0.02 \mathrm{~min}^{-1}, k_{2}=0.2 \mathrm{~min}^{-1}\), and \(k_{3}=2.0 \mathrm{~L} /...
Study smarter with the SolutionInn App