Simplifying Linear Algebra: Techniques, Solutions, and Examples

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

This question has to do with binary star systems, where 'i' is the angle of inclination of the system.
Calculate the mean expectation value of the factor ${\displaystyle {\sin }^3}$ i, i.e., the mean value it would have among an ensemble of binaries with random inclinations. Find the masses of the two stars, if ${\displaystyle {\sin }^3}$ i has its mean value.
Hint: In spherical coordinates, ${\displaystyle (\theta ,\varphi )}$, integrate over the solid angle of a sphere where the observer is in the direction of the z-axis, with each solid angle element weighted by ${\sin }^3\theta$.
${\displaystyle v_1=100k\frac{m}{s}}$
${\displaystyle v_2=200k\frac{m}{s}}$
Orbital period ${\displaystyle =2}$ days
${\displaystyle M_1=5.74e^{33}g}$
${\displaystyle M_2=2.87e^{33}g}$

The given matrix is the augmented matrix for a system of linear equations $[\begin{array}{}1& 0& -1& 0& -1& -2& 0\\ 0& 1& 2& 0& 1& 2& 0\\ 0& 0& 0& 1& 1& 1& 0\end{array}]$

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 ${\displaystyle x_1,x_2,\dots }$ from left to right.
$\left[\begin{array}{ccccc}1& 1& -3& 2& 1\\ 0& 1& 4& 0& 3\\ 0& 0& 0& 1& 2\end{array}\right]$

Systems of Inequalities Graph the solution set of the system if inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded.
$\{\begin{array}{l}y<9-x^2\\ y\ge x+3\end{array}$

Find or create an example of a system of equations with one solution. 
Graph and label the lines on a coordinate plane. Provide their equations. 
State the accurate solution to the system.

Below are various vectors in cartesian, cylindrical and spherical coordinates. Express the given vectors in two other coordinate systems outside the coordinate system in which they are expressed
$a)\overrightarrow{A}(x,y,z)={\overrightarrow{e}}_x$
$d)\overrightarrow{A}(\rho ,\varphi ,z)={\overrightarrow{e}}_{\rho }$
$g)\overrightarrow{A}(r,\theta ,\varphi )={\overrightarrow{e}}_{\theta }$
$j)\overrightarrow{A}(x,y,z)=\frac{-y{\overrightarrow{e}}_x+x{\overrightarrow{e}}_y}{x^2+y^2}$

Given point ${\displaystyle P(-2,6,3)}$ and vector ${\displaystyle B=y{a}_x+(x+z)a_y}$, Describe P and B in spherical and cylindrical coordinates. In the Cartesian, cylindrical, and spherical systems, evaluate A at P.

4.(a) Sketch the region of solutions of the following systems of inequalities:
${\displaystyle 2?+3?\le 12}$
${\displaystyle -?+2?<4}$
${\displaystyle ?\ge 0}$
${\displaystyle ?\ge 0}$
b. List the coordinates of all of the corner points:
c Which corner point would maximize the equation ${\displaystyle ?=5?+3??}$

Let $AX=B$ be a system of linear equations, where A is an $m\times nm\times n$ matrix, X is an n-vector, and $BB$ is an m-vector. Assume that there is one solution $X=X0$. Show that every solution is of the form $X0+Y$, where Y is a solution of the homogeneous system $AY=0$, and conversely any vector of the form $X0+Y$ is a solution.