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.

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2021-05-11
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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.

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avortarF

Answered 2021-05-12
Author has **113** answers

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 2021-05-12

Each of the matrices is the final matrix form for a system of two linear equations in the variables

asked 2022-05-23

At school, or in a first-year course on DEs, we learn (perhaps in less abstract language) that if you have a linear $n$th-order differential equation

$Ly=f$

then the general solution is something of the form

$y={a}_{1}{y}_{1}+...+{a}_{n}{y}_{n}+g$

where the ${y}_{i}$ are independent and satisfy $L{y}_{i}=0$, and $g$ satisfies $Lg=f$. Then we receive lots of training in how to find the ${y}_{i}$ and $g$.

Obviously any choice of the ${a}_{i}$ will give us a solution to $Ly=f$. But how do you know that there aren't any more solutions?

We justify this by making an analogy with systems of linear equations $Ax=b$, saying something along the lines of 'the space of solutions has the same dimension as the kernel of $A$'. But that works in finite dimensions - how do we know that the same is true with linear operators?

$Ly=f$

then the general solution is something of the form

$y={a}_{1}{y}_{1}+...+{a}_{n}{y}_{n}+g$

where the ${y}_{i}$ are independent and satisfy $L{y}_{i}=0$, and $g$ satisfies $Lg=f$. Then we receive lots of training in how to find the ${y}_{i}$ and $g$.

Obviously any choice of the ${a}_{i}$ will give us a solution to $Ly=f$. But how do you know that there aren't any more solutions?

We justify this by making an analogy with systems of linear equations $Ax=b$, saying something along the lines of 'the space of solutions has the same dimension as the kernel of $A$'. But that works in finite dimensions - how do we know that the same is true with linear operators?

asked 2021-09-29

Solve the system by clennaton

The solution is____

asked 2022-05-18

I have a non-linear form of the Poisson equation (with a diffusion coefficient that is a function of the derivatives of the dependent variable) that I'm trying to solve numerically (using FEM which requires the pde to be posed in its weak form).

$\mathrm{\nabla}\cdot (\eta \mathrm{\nabla}v)=G(x)$

where $\eta =\sqrt{1/2(\mathrm{\nabla}v:\mathrm{\nabla}v)}$ and $v=v(y,z)$. Multiplying by a test function θ and integrating wrt the domain, $\mathrm{\Omega}$, gives the weak form

${\oint}_{\mathrm{\Gamma}}\theta \cdot (\eta \mathrm{\nabla}v\cdot \mathbf{n})d\mathrm{\Gamma}-{\int}_{\mathrm{\Omega}}(\eta \mathrm{\nabla}v\cdot \mathrm{\nabla}\theta +G)d\mathrm{\Omega}=0$

However, none of my boundary conditions correspond to the Dirichelet type as I have ${\mathrm{\partial}}_{n}v=v$ on three of the boundaries in a rectangular domain and ${\mathrm{\partial}}_{n}v=0$ on the other (n is the normal vector to surface). Is this problem actually solvable using FEM, or at all? I have the solution to a simpler problem which might be a reasonable guess at the solution here.

$\mathrm{\nabla}\cdot (\eta \mathrm{\nabla}v)=G(x)$

where $\eta =\sqrt{1/2(\mathrm{\nabla}v:\mathrm{\nabla}v)}$ and $v=v(y,z)$. Multiplying by a test function θ and integrating wrt the domain, $\mathrm{\Omega}$, gives the weak form

${\oint}_{\mathrm{\Gamma}}\theta \cdot (\eta \mathrm{\nabla}v\cdot \mathbf{n})d\mathrm{\Gamma}-{\int}_{\mathrm{\Omega}}(\eta \mathrm{\nabla}v\cdot \mathrm{\nabla}\theta +G)d\mathrm{\Omega}=0$

However, none of my boundary conditions correspond to the Dirichelet type as I have ${\mathrm{\partial}}_{n}v=v$ on three of the boundaries in a rectangular domain and ${\mathrm{\partial}}_{n}v=0$ on the other (n is the normal vector to surface). Is this problem actually solvable using FEM, or at all? I have the solution to a simpler problem which might be a reasonable guess at the solution here.

asked 2021-05-07

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given row echelon form. Solve the system by back substitution. Assume that the variables are named x1,x2,

asked 2020-11-08

The given matrix is the augmented matrix for a system of linear equations.
Give the vector form for the general solution.

$\left[\begin{array}{cccc}1& 0& -1& -2\\ 0& 1& 2& 3\end{array}\right]$