# What is Linear system: Definition and 149 Discussions

In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be
modeled by linear systems.

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1. ### Solutions for nonrelativistic-matter perturbations

I'm studying the nonrelativistic-matter perturbations if the expansion of the Universe is driven by a combination of components. I'm currently Following this document (The growth of density perturbations) from Caltech. However, the author doesn't explain how he has found the solutions for the...
2. ### Solve the following linear system

Find the problem below with the solution indicated; The text approach is much clear to me... My way of tackling the problem is as follows; using echelon form (row reduction method) \begin{bmatrix} 1 & 1 & 2 & -5 &3\\ 2 & 5 & -1 & -9&-3 \\ 2 & 1 & -1 & 3&-11\\ 1& -3 & 2 & 7&-5...
3. ### Finding the general solution of this nonhomogeneous linear system

Hi, I was trying to do the following problem. My attempt. Finding the reduced row echelon form for the system above. I do not see any way to proceed any further. The following is the solution presented in solution manual. How do I proceed to get the following answer?
4. ### I Solving a complex linear system with parameters

I have to study the solutions of the following system of three equations and three unknowns upon variation of parameters k and h. ix1+kx2-x3 = 1+i (k+i)x1+(1-i)x2-(ik-1)x3 = h kx1+(4+2i)x2-(k-3-3i)x3 = 1-i Obviously i is the imaginary unit. And as stated k and h are the parameters . I can't...
5. ### MHB Linear system of equations: Echelon form/Solutions

Hey! 😊 I am looking at the following exercise but I think that I miss something. The statement is the following: We are given the following system of equations: \begin{align*}2a-2c+d-2e=&-2 \\ -2c-2d+2e=&\ \ \ \ \ 3 \\ d+2e=&-2\end{align*} 1) Is the system in echelon form? Justify. 2)...
6. ### My first proof ever - Linear algebra

First, a little context. It's been a while since I last posted here. I am a chemical engineer who is currently preparing for grad school, and I've been reviewing linear algebra and multivariable calculus for the last couple of months. I have always been successful at math (at least in the...
7. ### I This linear system has a solution? (Gauss' elimination)

I don't know the terms so I'm sorry if the informations at summary above is unclear. But I add a detailed photo of my calculations below. I use Gauss' Elimination laws.
8. ### How Do You Solve Systems of Linear Equations with Parameters?

1) x = 3 - 4p + q x = 3 - 4y + z x + 4y - z = 3 2) x + 4y - z = 3 (i) let x = a and y = b, so z = a + 4b - 3 General solution: x = a y = b z = a+ 4b - 3 (ii) let x = r and z = t, so y = (3 - r + t) / 4 General solution: x = r y = (3 - r + t) / 4 z = t3) I don't understand this part. Is the...
9. ### I Gaussian elimination for homogeneous linear systems

Hi, I ask for a clarification about the following: consider for instance a 10 x 12 homogeneous linear system and perform Gauss elimination for the first 8 unknowns. Suppose you end up with 5 equations in the remaining 12-8 = 4 unknowns (because in the process of the first 8 unknowns elimination...
10. ### MATLAB Ltiview and negative step response of a linear system

Hi, my professor gave me I code where he used to evaluate the answer of a linear system to a step increase in the input variables like this: MySystem = ss(A, B, C, D); ltiveiw('step', MySystem, 'r-', 300); My problem is that with this code I get the response only for a positive step. I'd...
11. ### B Can Linear Systems of Equations Have More Than Three Solutions?

According to my text, a linear system of equations is a problem described by two or more equations in two or more variables. Now the individual equations have infinitely many solutions, however, the system of equations is said to have either exactly one solution (one point of intersection...
12. ### Missing one equation to solve the linear system

Homework Statement So imagine 4 rigid rods connected together to form a dashed quadrilateral as shown in the picture. Now AB is fixed, can not be changed in anyway, while all other sides (AD, BC and CD) are connected but can move freely. The initial conditions (dashed quadrilateral) are given...
13. ### Discretize this continuous time linear system

Homework Statement I need to find the discrete time equivalent of the following system: \begin{bmatrix} \ddot{x} \\ \dot{x} \\ \ddot{\theta} \\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & 4.2042857 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 105.1071428 \\ 0 & 0 & 1 & 0 \end{bmatrix}...
14. ### Find values of k in this linear system

Homework Statement Find the values of k so that each of the following systems in unknowns x, y, and z has (i) a unique solution, (ii) no solution, (iii) an infinite number of solutions. x + y = kz = 1 x + ky + z = 1 kx + y + z = 1 Homework EquationsThe Attempt at a Solution I really don't...
15. M

### Constructing a 3x3 Linear system question

Homework Statement Construct a 3 × 3 example of a linear system that has 9 different coefficients on the left hand side but rows 2 and 3 become zero in elimination. If the right hand sude of your system is <b1,b2,b3> (Imagine this is a column vector) then how many solutions does your system...
16. M

### Finding values to make a linear system consistent

Homework Statement Given the following matrix: I need to determine the conditions for b1, b2, and b3 to make the system consistent. In addition, I need to check if the system is consistent when: a) b1 = 1, b2 = 1, b3 = 3 b) b1 = 1, b2 = 0., b3 = -1 c) b1 = 1, b2 = 2, b3 = 3 Homework...
17. ### [Linear System] What is the order of the system?

Hi, I plotted two Bode plots on MATLAB, and I'm wondering what's the system's order, because the magnitude plot looks like a first order (-20 dB/ dec) whether the phase plot looks like a second order (-180 degrees). I'm leaning towards a first-order system since the magnitude is more precise...
18. M

### MATLAB Solving Linear System with Eigenvalues in Matlab

Hi PF! I am looping through a linear system and each time I do I generate a new matrix, call this matrix ##A##. When finding the eigenvalues of ##A## in Matlab is use [a,sigma2M] = eig(A);% a eigenvector and sigma2M matrix of eigenvalues sigma2(:,ii) = sum(sigma2M);% create matrix with rows of...
19. ### MHB Which is the linear system of equations?

Hey! :o A national economy has $3$ sectors: fishing, forestry, boat building. One fishing boat is needed, to catch two tonnes of fish. To produce four tonnes of wood, one tonne of fish is needed to feed the forestry workers. Two tonnes of wood are needed, to build one fishing boat. There...
20. ### How Does Gaussian Elimination Solve This System of Equations?

Homework Statement system of equations is as follows x+y=1 2x+y-z=1 3x+y-2z=1 ##\begin{cases} x+y=1 |*(-1)\\2x+y-z=1\\3x+y-2z=1 \end{cases}## Homework Equations Gaussian elimination method technique The Attempt at a Solution ##\begin{cases} x+y=1 \\ x-z=0 |*(-2)\\ 2x-2z=0 \end{cases}##<=>...
21. ### Is the Integrator Linear in t?

Homework Statement Determine whether the system is linear Homework Equations Superposition The Attempt at a Solution I am comfortable solving the case where the bounds are from negative infinity to t. I have provided an example of that solution I found online. I attempt to solve that...
22. ### I Why do we linearise a non linear system

I am stuck with the concept of linearising the non-linear system...what i thought was...non-linear systems are defined by differential equations which is difficult to solve, hence we linearise them to make the differential equation look like a algebraic equation...am i right ?...or is there...
23. ### Solve System with Repeated Eigenvalues

Homework Statement I want to solve this systemx' = \left( \begin{array}\\ 7 & 1 \\ -4 & 3 \end{array} \right)x + \left( \begin{array}\\ t \\ 2t \end{array} \right) Homework EquationsThe Attempt at a Solution i found the eigenvalues to both be 5. The eigenvector is (1,-2) and the generalized...
24. ### Mathematica Mathematica: ill-conditioned linear system

Hi all, I have problem with regard to ill-conditioned linear system of solving sets of simultaneous equations using Mathematica program. I have tried my best to find a way to solve this but none was successful. I got results from m =1 and n =1 until m = 7 and n = 7, i,e. the systems are...
25. ### MHB Decide h so that the linear system has infinite solutions

Hi! I'm need some help with this question: Decide $h$ so that the linear system $Ax=b$ has infinite solutions. $$A=\pmatrix{ 5 & 6 & 7 \cr -7 & -4 & 1 \cr -4 & 4 & 16 \cr}$$ $$b=\pmatrix{ 6 \cr 30 \cr h \cr}$$ I solved a similar question before but with A being a 2x2 matrix (and B a 2x1) and...
26. ### How to plot the linear system solutions with multiple solutions?

Homework Statement Solve the linear system of equations: ax+by+z=1 x+aby+z=b x+by+az=1 for a,b\in\mathbb R and plot equations and solutions in cases where the system is consistent. Homework Equations -Cramer's rule -Kronecker-Capelli's theorem The Attempt at a Solution Using Cramer's rule, we...
27. ### Solution set: S = {(8 + 7z, 6 + 5z, z, 1) : z ∈ ℝ}

Homework Statement Plot the solution set of linear equations x-y+2z-t=1 2x-3y-z+t=-1 x+7z=8 and check if the set is a vector space. 2. The attempt at a solution Augmented matrix of the system: \begin{bmatrix} 1 & -1 & 2 & -1 & 1 \\ 2 & -3 & -1 & 1 & -1 \\ 1 & 0 & 7 & 0 & 8 \\...
28. ### Matrix-Vector Form Write an Augmented Matrix

Homework Statement Write in Vector-Matrix form then write the augmented matrix of the system. r + 2s + t = 1 r - 3s +3t = 1 4s - 5t = 3 Homework Equations The matrix to which the operations will be applied is called the augmented matrix of the system Ax = b, It is formed by appending the...
29. ### Linear system control with unmatched uncertainties

Hi everyone, I have a problem about linear system control with unmatched uncertainties, the system is \dot{e}=Ae+Bu+w where \dot means differentail sign, u is the control input, w is the disturbance, A is already Hurwitz, is there any way to design u such that the disturbance w can be offset...
30. ### MHB Solving Linear Systems with a Variable: How to Handle Unknown Constants?

Hello, I am stuck on a question for some time now and I am unsure how to solve this. I have tried the substitution and gaussian elimination methods but have had no luck at all. Identify the value(s) of k for which the following linear system x + 2y + z = 2 2x − 2y + 3z = 1 x + 3y + (k^2 − 3)z...
31. ### Linear System Solution Method and Validity

Homework Statement Find all solutions of the linear system x + 2y + 3z = a x + 3y + 8z = b x + 2y + 2z = c where a,b, and c are arbitrary constants. Homework EquationsThe Attempt at a Solution Using elimination, I managed to set the coefficients on the diagonal equal to 1, which then allowed...
32. ### MHB Finding unknown coefficients to give a unique solution in a linear system

Hi can someone please explain how to do this question: Given two equations; x-ay= 1 ax-4y=b For which values of a does each system have a unique solution, and for which pairs of values (a,b) does each system have more than one solution? All help is greatly appreciated
33. ### Linear System Solution in Matrix Form

\begin{cases} x+ 2y - z + w - t = 0 \\ x - y + z + 3w - 2t = 0 \end{cases} Add 1st to the 2nd: $$2x + y + w - t = 0 \\ y = -2x -w + t = 0$$ Substitute y in the 1st: ##x + 2x + w - t + 3w - 2t + z = 0 \\ z = 3x - 4w + 3t## Both z and y in terms of x,w,t. Writing using matrix form...
34. ### Find a basis of a linear system

Find a basis for the solution space of the linear system x1-x2-2x3+x4 = 0 -3x1+3x2+x3-x4 = 0 2x1-2x2+x3 = 0 I created a matrix (not augmented, will be 0 on right side no matter what row operations) and brought it to reduced echelon form. x2 and x4 were free variables and I set them to the...
35. ### RHS of Laplace's Equation is f(u(x,y))

Homework Statement I need to (computationally) solve the following linear elliptic problem for the function u(x,y): \Delta u(x,y) = u_{x,x} + u_{y,y} = k u(x,y) on the domain \Omega = [0,1]\times[0,1] with u(x,y) = 1 at all points on the boundary.Homework Equations [/B] I know that I...
36. ### Proving Equivalence of Iterative Refinement Equations for Linear Systems

Homework Statement The question relates to iterative refinement. The idea is that the computer generates a solution to the linear system Ax=b which is inexact (due to roundoff errors), denoted by x0. You then iterate the algorithm given in (1) until it converges to (something much closer to)...
37. ### Mathematical proof for non linear system

Homework Statement Consider the continuous-time processing system in figure 1, which has two inputs and one output. The linear sub-system H is characterised by the impulse response h(t) = e −2t , where t denotes time. Block diagram is the product of x1(t) and x2(t) going through a block step...
38. ### Infinitely Many Solutions: Proving or Disproving a Linear System

If a linear system has more unknowns than equations, then it must have infinitely many solutions. Prove or disprove. I'm not at all sure what to do with this one. Thank you very much for any help. :)
39. ### What Values of k Give Infinite Solutions in This Homogeneous System?

Homework Statement 1. Determine the values of k such that the following homogeneous linear system has infinitely number of solutions. My problem is: I cannot to find the value of k Homework Equations 2x – ky + z = 0 -x + y – 3kz = 0 kx – 2y + 2z = 0 The Attempt at a Solution After I...
40. ### Linear system - calculate output signal

Hi! Homework Statement a) Calculate and sketch amplitude spectrum of u(t), b) u(t) is input signal for linear time invariant system whose transfer function H(jw) is shown. Calculate output signal uo(t) Homework EquationsThe Attempt at a Solution I completed task a), I got...
41. ### Inhomogeneous Linear System Solutions

Hi all, I have been searching around and cannot seem to find an answer. I am doing a past paper, and have the answers but do not understand one part. I hope someone can help. The question is: A \in M_{3 \times 3}(\mathbb{R}), \vec{b} \in \mathbb{R}^{3}. The general solution to the equation...
42. ### Linear system of differential equations

Hi! My task is to solve this system: $$\frac{\mathrm{d} x}{\mathrm{d} t}=-x+y-2z$$ $$\frac{\mathrm{d} y}{\mathrm{d} t}=4x+y$$ $$\frac{\mathrm{d} z}{\mathrm{d} t}=2x+y-z$$ My first equation (1) is $$\frac{\mathrm{d} y}{\mathrm{d} t}=4x+y$$. Derivative of (1) is \frac{\mathrm{d} }{\mathrm{d}...
43. ### Higher order D.E. to linear system of 1st order D.E.'s

Homework Statement \textbf{(a)} This is an exercise from a course on numerical analysis. Write the system of differential equations u''' = x^2uu'' - uv' v'' = xvv' + 4u' as a first order system of differential equations, \textbf{y'} = \textbf{y}(x,\textbf{y}). \textbf{(b)} Determine...
44. ### MHB What are the values of α for any initial value of x(0) in a linear system?

Considering the following find the values for α for any initial value of x(0). Any help will be useful, thank you!
45. ### Linear system of equations with non-trivial solutions

Homework Statement Let (**) \begin{matrix} a_{11}x_1 & + & \ldots & + & a_{1n}x_n & = 0 \\ \vdots & & & & \vdots\\ a_{m1}x_1 & + & \ldots & + & a_{mn}x_n & = 0 \\ \end{matrix} be a system of m linear equations in n unknowns, and...
46. ### MHB Linear approximation of Non linear system by Taylor series

I have a equation which represents a nonlinear system.I need to linearize it to obtain a linear system.I have studied various notes and asked my teachers but they are unable to explain how the solution has been obtained.I have the solution but I want to know how it has been done.Please could...
47. ### Linear System Stability Analysis: Matrix Blocks, Integral Block Included

Homework Statement I'm having to figure out if a system is asymptotically stable, stable, or unstable. I am given the system block diagram. However, each constant block is actually a matrix. Also, there is an integral block thrown in there... Homework Equations The Attempt at a Solution In a...