Find the Equivalent Polar Equation for a given Equation with Rectangular Coordinates: r cos theta= -1

Consider the linear system
${\overrightarrow{y}}^{\prime }=\left[\begin{array}{cc}-3& -2\\ 6& 4\end{array}\right]\overrightarrow{y}$
a) Find the eigenvalues and eigenvectors for the coefficient matrix
${\lambda }_{1}1,{\overrightarrow{v}}_1=\left[\begin{array}{c}-1\\ 2\end{array}\right]$ and ${\lambda }_2=0,{\overrightarrow{v}}_2=\left[\begin{array}{c}-2\\ 3\end{array}\right]$
b) For each eigenpair in the previos part, form a solution of ${\displaystyle {\overrightarrow{y}}^{\prime }=A\overrightarrow{y}}$. Use t as the independent variable in your answers.
${\overrightarrow{y}}_1(t)=\left[\begin{array}{cc}& \\ & \end{array}\right]$ and ${\overrightarrow{y}}_2(t)=\left[\begin{array}{c}-2\\ 3\end{array}\right]$
c) Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solution? No, it is not a fundamental set.

Find linear transformation matrix from a linear transformation matrix of its composition
Let's say that $V$ is a vector space over the field $K$ and suppose we have a linear transformation $f\in \text{End}V$ thats matrix is know on some basis.
How to find a matrix of a linear transformation $g\in \text{End}V$ on the same basis, so that $g\circ g=f$.
For example if $V$ is vector space over the field ${\mathbb{Z}}_{11}$ and matrix of linear transformation $f\in \text{End}V$ on some basis is
$\left(\begin{array}{cccc}\overline{7}& \overline{3}& \overline{4}& \overline{0}\\ \overline{0}& \overline{6}& \overline{9}& \overline{6}\\ \overline{1}& \overline{9}& \overline{3}& \overline{1}\\ \overline{0}& \overline{2}& \overline{8}& \overline{5}\end{array}\right)$
Find matrix of a linear transformation $g\in \text{End}V$ on the same basis, so that $g\circ g=f$

What is the transformation matrix?
In ${\mathbb{R}}^{\mathbb{2}}$ a basis is given $a=(a_1,a_2)$ where:
$a_1=(1,-1)$
$a_2=(0,1)$
For $f:{\mathbb{R}}^{\mathbb{2}}\to {\mathbb{R}}^{\mathbb{2}}$ it is known:
$f(a_1)=-6\cdot a_1$
$f(a_2)=1\cdot a_2$
Determine the matrix of transformation with regards to the standard e-basis.

Find a set of vectors that spans the plane ${\displaystyle P1:2x+3y-z=0}$

Since $\sqrt{2}\approx 1+\frac{1}{1+1+\frac{1}{1+1+\frac{1}{1+\sqrt{2}}}}$ I thought i could do $\frac{r_{n+1}}{s_{n+1}}=1+\frac{1}{1+1+\frac{1}{1+1+\frac{1}{1+\frac{r_n}{s_n}}}}$ with the following piece of code :

ite = 8
int r = 1;
int s = 1;
int R = 0;
for (int i=0,i<ite,1){
    R = 3*r+4*s;
    s=2*r+3*s;
    r=R
}
int result = r/s;
System.out.println(result);

But it's not exactly the code one would get using picard's iteration method, so my answer does not fit the criteria. What am-I doing wrong?

 

If one does not know the answer but knows how to visualise picard's method for calculating sqrt(2) on a graph that would already help me a lot.

 

Thanks in advance

Let $A$ be the set of all $n\times n$ symmetric real matirix and $f\in C(\mathbb{R},A)$. Then whether there is a $g\in C(\mathbb{R},O(n))$ such that for all $t\in \mathbb{R}$, $g(t{)}^{-1}f(t)g(t)$ is a diagonal matrix?

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.