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.
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...
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...
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?
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...
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)...
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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}##<=>...
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...
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...
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...
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...
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...
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...
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 \\...
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...
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...
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...
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...
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
\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...
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...
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...
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)...
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...
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. :)
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...
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...
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...
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}...
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...
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...
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...
Hellow! I have searched for some theory about linear system in polar coordinates, unfortunately, I not found anything... exist some theory, some book, anything about this topic for study? Thanks!
Homework Statement
The unique solution of the linear system of differential equations
##\frac{dv}{dt}=-34v+ -16w, v(0)=-1##
##\frac{dw}{dt}=80v+ 38w, w(0)=-3##
is: (enter the smaller of the eigenvalues first, and note that all entries here are integers)
##v(t)= C_1 e^{-2t}+C_2 e^{6t}##...