Boost Your Understanding of Linear Equation Forms

Write the vector form of the general solution of the given system of linear equations.
x1+4x4=0
x2−2x4 ​

In the matrix of a system of linear equations, suppose that two of the rows are equal. What can you say about the rowreduced form of the matrix?

$x+y+2z=3$
$3x+2y+2z=4$
$x+y+3z=5$

Determine whether the statement is true or false. If the last row of the reduced row echelon form of the augmented matrix of a system of linear equations has only one nonzero entry, then the system is inconsistent.

When solving a system of linear equations using matrices, do you prefer to place the augmented matrix in row echelon form or in reduced row echelon form? Give reasons for your choice.

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $x_1,x_2,...$ form left to right. $\left[\begin{array}{ccccc}1& 0& 5& 3& 2\\ 0& 1& -2& 4& -7\\ 0& 0& 1& 1& 3\end{array}\right]$

The reduced row echelon form of the augmented matrix of a system of three linear equations in three variables must be of the form

$\left[\begin{array}{cccc}1& 0& 0& a\\ 0& 1& 0& b\\ 0& 0& 1& c\end{array}\right]$

where a, b, c, are real numbers. Justify your answer.

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent.

$\left[\begin{array}{ccccc}1& 0& 0& |& -1\\ 0& 1& 0& |& 3\\ 0& 0& 1& |& 4\\ 0& 0& 0& |& 0\end{array}\right]$

find an equation of the form $y=a{x}^2+bx+c$, whose graph passess through the points (2, 3), (-2, 7) and (1, -2). You must use a system of linear equations in three variables for this problem.

The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.
${\displaystyle {\lambda }_1=1\Rightarrow \left\{\left[\begin{array}{c}2\\ -1\end{array}\right]\right\},{\lambda }_2=3\Rightarrow \left\{\left[\begin{array}{c}3\\ 1\end{array}\right]\right\}}$

Solve the system of linear equations using the substitution method.
$x+y-z=4$

$3x+2y+4z=17$

$-x+5y+z=8$

The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.
${\displaystyle \lambda 1=-1\to \left\{\begin{array}{cc}1& 1\end{array}\right\},\lambda 2=2\to \left\{\begin{array}{cc}1& -1\end{array}\right\}}$

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent. $\left[\begin{array}{ccccc}1& 0& -1& |& 1\\ 0& 1& 2& |& 1\end{array}\right]$

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $x_1,x_2\dots$ from left to right. $[1,2,0,2,-1,3]$

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use x, y; or x, y, z; or x1,x2,x3,x4x as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. ⎡⎣⎢100010000002⎤⎦⎥

Let [A[A b]b] be the augmented matrix of a system of linear equations. Prove that if its reduced row echelon form is [R[R c]c], then R is the reduced row echelon form of A.

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 $x_1-6x_2+x_3=5$.
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

Each of the matrices is the final matrix form for a system of two linear equations in the variables x1 and x2. Write the solution of the system.
$\left[\begin{array}{cccc}1& -2& |& 7\\ 0& 0& |& -9\end{array}\right]$

Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has an infinite number of solutions.

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x,y;x,y;orx,y,z;x,y,z;or{x}_1,x_2,x_3,x_4$ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.

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