tabita57i
2021-03-04
Answered

Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations.

$\{\begin{array}{c}x+y=0\\ 5x-2y-2z=12\\ 2x+4y+z=5\end{array}$

You can still ask an expert for help

Cristiano Sears

Answered 2021-03-05
Author has **96** answers

For a system of equations $\{\begin{array}{c}ax+by+cz=j\\ dx+ey+fz=k\\ gx+hy+iz=l\end{array}$

, the coefficient matrix is$\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$ and the augmented matrix is $\left[\begin{array}{ccccc}a& b& c& |& j\\ d& e& f& |& k\\ g& h& i& |& l\end{array}\right]$

a) For the system$\{\begin{array}{c}x+y=0\\ 5x-2y-2z=12\\ 2x+4y+z=5\end{array}$

a=1 , b=1 , c=0,d=5,e=-2, f=-2, g=2 , h=4 and i=1 so the coefficient matrix is$\left[\begin{array}{ccc}5& 1& 0\\ 5& -2& -2\\ 2& 4& 1\end{array}\right]$

b) For the system$\{\begin{array}{c}x+y=0\\ 5x-2y-2z=12\\ 2x+4y+z=5\end{array}$

a=1 , b=1 , c=0,d=5,e=-2, f=-2, g=2 , h=4 , i=1 , j=0 , k=12 and l=5 so the augmented matrix is$\left[\begin{array}{ccccc}5& 1& 0& |& 0\\ 5& -2& -2& |& 12\\ 2& 4& 1& |& 5\end{array}\right]$

Answer

a)$\left[\begin{array}{ccc}5& 1& 0\\ 5& -2& -2\\ 2& 4& 1\end{array}\right]$

b)$\left[\begin{array}{ccccc}5& 1& 0& |& 0\\ 5& -2& -2& |& 12\\ 2& 4& 1& |& 5\end{array}\right]$

, the coefficient matrix is

a) For the system

a=1 , b=1 , c=0,d=5,e=-2, f=-2, g=2 , h=4 and i=1 so the coefficient matrix is

b) For the system

a=1 , b=1 , c=0,d=5,e=-2, f=-2, g=2 , h=4 , i=1 , j=0 , k=12 and l=5 so the augmented matrix is

Answer

a)

b)

asked 2021-06-01

Find the linear approximation of the function

Use L(x) to approximate the numbers

asked 2022-02-23

Write the following linear differential equations with constant coefficients in the form of the linear system

Hint: Let

I have tried to do this in the following way but I do not know if I am doing well:

Let

and thus

asked 2022-05-16

Consider the following linear system of differential equations:

$\{\begin{array}{l}\dot{x}=-4y\\ \dot{y}=x\end{array}$

where $x(t)$ and $y(t)$ are unknown real functions.

One can simply verify that the general solution is

$\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)={c}_{1}\left(\begin{array}{c}\mathrm{cos}(2t)\\ \frac{1}{2}\mathrm{sin}(2t)\end{array}\right)+{c}_{2}\left(\begin{array}{c}-2\mathrm{sin}(2t)\\ \mathrm{cos}(2t)\end{array}\right)$

where ${c}_{1}$ and ${c}_{2}$ are real parameters.

Question: Which is the exponential form of this expression? It should by something like

$\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)={k}_{1}{e}^{i2t}\left(\begin{array}{c}{P}_{11}\\ {P}_{21}\end{array}\right)+{k}_{2}{e}^{-i2t}\left(\begin{array}{c}{P}_{12}\\ {P}_{22}\end{array}\right)$

where ${k}_{1}$ and ${k}_{2}$ should (?) be complex parameters.

$\{\begin{array}{l}\dot{x}=-4y\\ \dot{y}=x\end{array}$

where $x(t)$ and $y(t)$ are unknown real functions.

One can simply verify that the general solution is

$\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)={c}_{1}\left(\begin{array}{c}\mathrm{cos}(2t)\\ \frac{1}{2}\mathrm{sin}(2t)\end{array}\right)+{c}_{2}\left(\begin{array}{c}-2\mathrm{sin}(2t)\\ \mathrm{cos}(2t)\end{array}\right)$

where ${c}_{1}$ and ${c}_{2}$ are real parameters.

Question: Which is the exponential form of this expression? It should by something like

$\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)={k}_{1}{e}^{i2t}\left(\begin{array}{c}{P}_{11}\\ {P}_{21}\end{array}\right)+{k}_{2}{e}^{-i2t}\left(\begin{array}{c}{P}_{12}\\ {P}_{22}\end{array}\right)$

where ${k}_{1}$ and ${k}_{2}$ should (?) be complex parameters.

asked 2022-02-22

Obtain just a particular solution (the general solution can alwaysbe obtained easily by adding an arbitrary multiple of a solution of the associated homogeneous equation). In these exercises, the forcing function is an elementary sinusoidal function. If the forcing function is $\mathrm{cos}5t$ , then a particular solution of the form A $\mathrm{cos}5t$ will not work. However, a linear combination of $\mathrm{cos}5t$ and $\mathrm{sin}5t$ will work. Thus, the form for xp is $A\mathrm{cos}5t+B\mathrm{sin}5t$ , where the constants A and B are determined by substituting the assumed form of the particular solution into the differential equation for x. In Exercises 47–52, the forcing function has several different terms. Use a form for xp consisting of the sum of the forms for each term.

$\frac{dx}{dt}+x=2{e}^{3t}+\mathrm{sin}t$

asked 2022-02-25

I mean, that whenever we talk about linear equations, we say that they always form a straight line on a graph. We make this sure by making a graph of some possible solutions to the equation and plotting the points on the graph. But can we make sure that all the solutions of the linear equation will always form a straight line because we cannot plot each and every solution of the equation on the graph because there are infinitely many solutions to a linear equation in 2 variables?

asked 2022-02-17

How to solve the system of linear differential equation of the form

${x}^{\prime}=Ax+b$

I can solve the homogeneous form by finding the eigenvalues and respective eigenvectors, but how to find the particular solution part. Also is there any limitation from getting eigenvalues positive, negative or complex. Any other different method involving matrix algebra is also welcome.

I can solve the homogeneous form by finding the eigenvalues and respective eigenvectors, but how to find the particular solution part. Also is there any limitation from getting eigenvalues positive, negative or complex. Any other different method involving matrix algebra is also welcome.

asked 2022-02-16

The linear equation is the equation that has a from:

${a}_{1}{x}_{1}+{a}_{2}{x}_{2}+\dots +{a}_{n}{x}_{n}=b$

But is this really the definition of a linear equation? I thought that the definition should be the form of linear mapping. Like,

$f(x+y)=f\left(x\right)+f\left(y\right)$

$f\left(ax\right)=af\left(x\right)$

What is the definition of a linear equation?

But is this really the definition of a linear equation? I thought that the definition should be the form of linear mapping. Like,

What is the definition of a linear equation?