To solve:the system of equation

Give a correct answer for given question

(A) Argue why $(1,0,3),(2,3,1),(0,0,1)$ is a coordinate system ( bases ) for $R^3$ ?

(B) Find the coordinates of $(7,6,16)$ relative to the set in part (A)

Consider the following vectors in ${\displaystyle R^4:}$ $v_1=\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right],v_2=\left[\begin{array}{c}0\\ 1\\ 1\\ 1\end{array}\right]v_3=\left[\begin{array}{c}0\\ 0\\ 1\\ 1\end{array}\right],v_4=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]$ d. If $x=\left[\begin{array}{c}23\\ 12\\ 10\\ 19\end{array}\right],find{\left\{x\right\}}_{B}e$. If $x_B=\left[\begin{array}{c}3\\ 1\\ -4\\ -4\end{array}\right]$, find x.

Let ${\displaystyle \gamma =\{t^2-t+1,t+1,t^2+1\}\text{and}\beta =\{t^2+t+4,4t^2-3t+2,2t^2+3\}be\text{or}deredbasesf\text{or}{P}_2\left(R\right).}$ Find the change of coordinate matrix Q that changes $\beta \text{ coordinates into }\gamma -\text{ coordinates}$

The given system of inequality:
${\displaystyle \{\begin{array}{c}y<9-x^2\\ y\ge x+3\end{array}}$
Also find the coordinates of all vertices, and check whether the solution set is bounded.

How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates?

Prove that by setting these two expressions equal to another, the result is an identity:
${\displaystyle d={\cos }^{-1}(\sin \left(L{T}_1\right)\sin \left(L{T}_2\right)+\cos \left(L{T}_1\right)\cos \left(L{T}_2\right)\cos \left({\ln }_1-{\ln }_2\right))}$
${\displaystyle d=2{\sin }^{-1}\left(\sqrt{{\sin }^2\left(\frac{L{T}_1-L{T}_2}{2}\right)+\cos \left(L{T}_1\right)\cos \left(L{T}_2\right){\sin }^2\left(\frac{{\ln }_1-{\ln }_2}{2}\right)}\right)}$

Find the value of x or y so that the line passing through the given points has the given slope. (9, 3), (-6, 7y), $m=3$