Simplifying Linear Algebra: Techniques, Solutions, and Examples

Give an example of the row-echelon form of an augmented matrix that corresponds to an infinitely solvable system of linear equations.

Each of the matrices is the final matrix form for a system of two linear equations in the variables $x_1$ and $x_2$. Write the solution of the system.

$\left[\begin{array}{cccc}1& 3& |& 2\\ 0& 0& |& 4\end{array}\right]$

Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations. $\{\begin{array}{l}7x-5y=11\\ x-y=-5\end{array}$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A system of linear equations in three variables, x, y, and z cannot contain an equation in the form $y=mx+b$.

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.

$\left[\begin{array}{ccc}1& 0& -6\\ 1& 1& 3\\ 0& 0& 0\end{array}\right]$

Substitute each point (-3, 5) and (2, -1) into the slope-intercept form of a linear equation to write a system of equations. Then use the system to find the equation of the line containing the two points.

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,$\dots$ from left to right.
$\left[\begin{array}{cccc}1& -3& 7& 1\\ 0& 1& 4& 0\\ 0& 0& 0& 1\end{array}\right]$

Write the given system of linear equations as a matrix equation of the form Ax=b.
$x_1-2x_2+3x_3=0$
$2x_1+x_2-5x_3=4$

Vectors u and v are orthogonal. If $u=⟨3,1+b⟩$ and $v=⟨5,1-b⟩$ find all possible values for b.

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. $\left[\begin{array}{cccccc}1& 2& 0& 0& |& -4\\ 0& 0& 1& 0& |& -3\\ 0& 0& 0& 1& |& 2\\ 0& 0& 0& 0& |& 0\end{array}\right]$

Each of the matrices is the final matrix form for a system of two linear equations in the variables $x_1$ and $x_2$. Write the solution of the system. $\left[\begin{array}{cccc}1& 5& |& 10\\ 0& 0& |& 0\end{array}\right]$

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.
$\lambda 1=2\Rightarrow \{[4,3,1]\},\lambda 2=-2\Rightarrow \{[1,2,0],[2,3,1]\}$

Identify the surface with the given vector equation. $r(s,t)=$$(s,t,t^2-s^2)$
eliptic cylinder
circular paraboloid
hyperbolic paraboloid
plane
circular cylinder