If an echelon form of an augmented matrix for a linear system of equations has a row of the form ([0…0 1| 0]

, then the system has no solution

True or false (justify)

conyezere
2022-06-09

If an echelon form of an augmented matrix for a linear system of equations has a row of the form ([0…0 1| 0]

, then the system has no solution

True or false (justify)

You can still ask an expert for help

asked 2021-06-10

Determine whether the given set S is a subspace of the vector space V.

A. V=${P}_{5}$ , and S is the subset of ${P}_{5}$ consisting of those polynomials satisfying p(1)>p(0).

B.$V={R}_{3}$ , and S is the set of vectors $({x}_{1},{x}_{2},{x}_{3})$ in V satisfying ${x}_{1}-6{x}_{2}+{x}_{3}=5$ .

C.$V={R}^{n}$ , and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.

D. V=${C}^{2}(I)$ , and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.

E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.

F. V=${P}_{n}$ , and S is the subset of ${P}_{n}$ consisting of those polynomials satisfying p(0)=0.

G.$V={M}_{n}(R)$ , and S is the subset of all symmetric matrices

A. V=

B.

C.

D. V=

E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.

F. V=

G.

asked 2022-06-25

The problem reads:

The solution of a certain differential equation is of the form

$y(t)=a\mathrm{exp}(5t)+b\mathrm{exp}(8t)$

where $a$ and $b$ are constants.

The solution has initial conditions $y(0)=5$ and ${y}^{\prime}(0)=5$

Find the solution by using the initial conditions to get linear equations for $a$ and $b$. ....................

What I did was solve using the initial conditions and I found that

$a+b=5$

and $5a+8b=5.$

Am I totally on the wrong track? I don't know what it means to find a linear equation for $a$ and $b$. I'd appreciate it if you could solve it step by step.

The solution of a certain differential equation is of the form

$y(t)=a\mathrm{exp}(5t)+b\mathrm{exp}(8t)$

where $a$ and $b$ are constants.

The solution has initial conditions $y(0)=5$ and ${y}^{\prime}(0)=5$

Find the solution by using the initial conditions to get linear equations for $a$ and $b$. ....................

What I did was solve using the initial conditions and I found that

$a+b=5$

and $5a+8b=5.$

Am I totally on the wrong track? I don't know what it means to find a linear equation for $a$ and $b$. I'd appreciate it if you could solve it step by step.

asked 2022-04-29

3.1 where 1 is repeating

asked 2022-04-10

I am trying to interpolate a function defined over a three-dimensional real space:

$f:{R}^{3}\to R\phantom{\rule{0ex}{0ex}}(x,y,z)\to f(x,y,z)$

Let assume I have ${N}_{1}{N}_{2}{N}_{3}$ points in the space which form my grid for this interpolation, and the multivariate series

$F(x,y,z)=\sum _{a=0}^{{N}_{1}-1}\sum _{b=0}^{{N}_{2}-1}\sum _{c=0}^{{N}_{3}-1}{C}_{abc}{x}^{a}{y}^{b}{z}^{c}$

is the chosen interpolator. In order to find the coefficients I should form the following sets of equations:

$\sum _{a=0}^{{N}_{1}-1}\sum _{b=0}^{{N}_{2}-1}\sum _{c=0}^{{N}_{3}-1}{C}_{abc}{x}_{i}^{a}{y}_{i}^{b}{z}_{i}^{c}={f}_{i}=f({x}_{i},{y}_{i},{z}_{i})\phantom{\rule{thinmathspace}{0ex}}{\textstyle \text{, where,}}i=1,\dots ,{N}_{1}{N}_{2}{N}_{3}\phantom{\rule{thinmathspace}{0ex}},$

and then solve for the coefficients. However, for this I first need to write the above set of linear equations in the standard form

$A\overrightarrow{x}=\overrightarrow{b}$

wherein, A is the matrix of coefficients, $\overrightarrow{x}$ is the vector of unknowns, and $b=\{{f}_{i}{\}}_{1}^{{N}_{1}{N}_{2}{N}_{3}}$ is the known vector. For this to be done I would require to expand the multivariate power series in the form of a single variable series, that is,

$\sum _{a=0}^{{N}_{1}-1}\sum _{b=0}^{{N}_{2}-1}\sum _{c=0}^{{N}_{3}-1}{C}_{abc}{x}_{i}^{a}{y}_{i}^{b}{z}_{i}^{c}=\sum _{m=0}^{({N}_{1}-1)({N}_{2}-1)({N}_{3}-1)}{C}_{m}{\eta}_{i}^{m}$

wherein ${\eta}_{i}={\eta}_{i}({x}_{i},{y}_{i},{z}_{i})$ and probably $m=abc$. OF course the expansion needs to be nontrivial and useful.

Is it possible at all? Any suggestion to find the coefficients more practically?

Regards, owari

UPDATE.

Maybe it appears that the most natural way for solving this problem is comprised of the following steps:

1. first solve for the coefficients of

$\sum _{a=0}^{{N}_{1}-1}{E}_{a}(y,z){x}_{i}^{a}=f({x}_{i},y,z)\phantom{\rule{thinmathspace}{0ex}}{\textstyle \text{, where,}}i=1,\dots ,{N}_{1}\phantom{\rule{thinmathspace}{0ex}},$

2. then solve for the coefficients of

$\sum _{b=0}^{{N}_{2}-1}{D}_{ab}(z){y}_{i}^{b}={E}_{a}({y}_{i},z)\phantom{\rule{thinmathspace}{0ex}}{\textstyle \text{, where,}}i=1,\dots ,{N}_{2}\phantom{\rule{thinmathspace}{0ex}}{\textstyle \text{, and,}}a=0,\dots ,{N}_{1}-1$

3. and finally solve for the coefficients of

$\sum _{c=0}^{{N}_{2}-1}{C}_{abc}{z}_{i}^{c}={D}_{ab}({z}_{i})\phantom{\rule{thinmathspace}{0ex}}{\textstyle \text{, where,}}i=1,\dots ,{N}_{3}\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}a=0,\dots ,{N}_{1}-1{\textstyle \text{, and,}}b=0,\dots ,{N}_{2}-1$

which gives the coefficients we were looking for. However, this way, the first two steps will be working with function-type coefficients instead of numerical coefficients and that will prevent efficient usage of the available codes in numerical analysis. Solving for each set of coefficients at each step for different grid points would also inevitably increase the number of equations drastically, so any better suggestion to solve for this problem?

$f:{R}^{3}\to R\phantom{\rule{0ex}{0ex}}(x,y,z)\to f(x,y,z)$

Let assume I have ${N}_{1}{N}_{2}{N}_{3}$ points in the space which form my grid for this interpolation, and the multivariate series

$F(x,y,z)=\sum _{a=0}^{{N}_{1}-1}\sum _{b=0}^{{N}_{2}-1}\sum _{c=0}^{{N}_{3}-1}{C}_{abc}{x}^{a}{y}^{b}{z}^{c}$

is the chosen interpolator. In order to find the coefficients I should form the following sets of equations:

$\sum _{a=0}^{{N}_{1}-1}\sum _{b=0}^{{N}_{2}-1}\sum _{c=0}^{{N}_{3}-1}{C}_{abc}{x}_{i}^{a}{y}_{i}^{b}{z}_{i}^{c}={f}_{i}=f({x}_{i},{y}_{i},{z}_{i})\phantom{\rule{thinmathspace}{0ex}}{\textstyle \text{, where,}}i=1,\dots ,{N}_{1}{N}_{2}{N}_{3}\phantom{\rule{thinmathspace}{0ex}},$

and then solve for the coefficients. However, for this I first need to write the above set of linear equations in the standard form

$A\overrightarrow{x}=\overrightarrow{b}$

wherein, A is the matrix of coefficients, $\overrightarrow{x}$ is the vector of unknowns, and $b=\{{f}_{i}{\}}_{1}^{{N}_{1}{N}_{2}{N}_{3}}$ is the known vector. For this to be done I would require to expand the multivariate power series in the form of a single variable series, that is,

$\sum _{a=0}^{{N}_{1}-1}\sum _{b=0}^{{N}_{2}-1}\sum _{c=0}^{{N}_{3}-1}{C}_{abc}{x}_{i}^{a}{y}_{i}^{b}{z}_{i}^{c}=\sum _{m=0}^{({N}_{1}-1)({N}_{2}-1)({N}_{3}-1)}{C}_{m}{\eta}_{i}^{m}$

wherein ${\eta}_{i}={\eta}_{i}({x}_{i},{y}_{i},{z}_{i})$ and probably $m=abc$. OF course the expansion needs to be nontrivial and useful.

Is it possible at all? Any suggestion to find the coefficients more practically?

Regards, owari

UPDATE.

Maybe it appears that the most natural way for solving this problem is comprised of the following steps:

1. first solve for the coefficients of

$\sum _{a=0}^{{N}_{1}-1}{E}_{a}(y,z){x}_{i}^{a}=f({x}_{i},y,z)\phantom{\rule{thinmathspace}{0ex}}{\textstyle \text{, where,}}i=1,\dots ,{N}_{1}\phantom{\rule{thinmathspace}{0ex}},$

2. then solve for the coefficients of

$\sum _{b=0}^{{N}_{2}-1}{D}_{ab}(z){y}_{i}^{b}={E}_{a}({y}_{i},z)\phantom{\rule{thinmathspace}{0ex}}{\textstyle \text{, where,}}i=1,\dots ,{N}_{2}\phantom{\rule{thinmathspace}{0ex}}{\textstyle \text{, and,}}a=0,\dots ,{N}_{1}-1$

3. and finally solve for the coefficients of

$\sum _{c=0}^{{N}_{2}-1}{C}_{abc}{z}_{i}^{c}={D}_{ab}({z}_{i})\phantom{\rule{thinmathspace}{0ex}}{\textstyle \text{, where,}}i=1,\dots ,{N}_{3}\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}a=0,\dots ,{N}_{1}-1{\textstyle \text{, and,}}b=0,\dots ,{N}_{2}-1$

which gives the coefficients we were looking for. However, this way, the first two steps will be working with function-type coefficients instead of numerical coefficients and that will prevent efficient usage of the available codes in numerical analysis. Solving for each set of coefficients at each step for different grid points would also inevitably increase the number of equations drastically, so any better suggestion to solve for this problem?

asked 2022-05-18

I am studying about the linear odes with non-constant coefficients.

I know the first order linear ode with non-constant coefficient

$\begin{array}{}\text{(1)}& {y}^{{}^{\prime}}(x)+f(x)y(x)=0\end{array}$

has a general solution of the form

$\begin{array}{}\text{(2)}& y=C{e}^{-\int f(x)dx}\end{array}$

However, I am more interested in the case of linear second order odes with non-constant coefficients

$\begin{array}{}\text{(3)}& {y}^{{}^{\u2033}}(x)+g(x){y}^{{}^{\prime}}(x)+f(x)y(x)=0\end{array}$

I know that this equation does not have a closed form solution like (2). However, I am interested in special cases of that.

Questions

1. Consider (3), when $g(x)=0$, then we have

$\begin{array}{}\text{(4)}& {y}^{{}^{\u2033}}(x)+f(x)y(x)=0\end{array}$

Is Eq.(4) a famous well-known equation? If YES, what is its name?

2. Does (4) have a closed form solution like (2)?

3. Can you name or give me a list of well-known linear second order odes with non-constant coefficients which are not polynomial?

For example, I know Cauchy-Euler, Airy, Bessel, Chebyshev, Laguerre and Legendre equations whose coefficients are polynomials. But I don't know any well-known equation with non-polynomial coefficients.

I know the first order linear ode with non-constant coefficient

$\begin{array}{}\text{(1)}& {y}^{{}^{\prime}}(x)+f(x)y(x)=0\end{array}$

has a general solution of the form

$\begin{array}{}\text{(2)}& y=C{e}^{-\int f(x)dx}\end{array}$

However, I am more interested in the case of linear second order odes with non-constant coefficients

$\begin{array}{}\text{(3)}& {y}^{{}^{\u2033}}(x)+g(x){y}^{{}^{\prime}}(x)+f(x)y(x)=0\end{array}$

I know that this equation does not have a closed form solution like (2). However, I am interested in special cases of that.

Questions

1. Consider (3), when $g(x)=0$, then we have

$\begin{array}{}\text{(4)}& {y}^{{}^{\u2033}}(x)+f(x)y(x)=0\end{array}$

Is Eq.(4) a famous well-known equation? If YES, what is its name?

2. Does (4) have a closed form solution like (2)?

3. Can you name or give me a list of well-known linear second order odes with non-constant coefficients which are not polynomial?

For example, I know Cauchy-Euler, Airy, Bessel, Chebyshev, Laguerre and Legendre equations whose coefficients are polynomials. But I don't know any well-known equation with non-polynomial coefficients.

asked 2021-05-11

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution.

asked 2021-05-18

The row echelon form of a system of linear equations is given.

(a)Write the system of equations corresponding to the given matrix. Use x, y; or x, y, z; or

(b)Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.