Understanding Coordinate Systems and Equations

To graph: The sketch of the solution set of system of nonlinear inequality
${\displaystyle [x^2+y^2\ge 9]}$
${\displaystyle x^2+y^2\le 25}$
${\displaystyle y\ge \left|x\right|}$

Interraption: To show that the system ${\displaystyle \dot{r}=r(1-r^2),\dot{\theta }=1}$ is equivalent to ${\displaystyle \dot{x}=x-y-x(x^2+y^2),\dot{y}=x+y-y(x^2+y^2)}$ for polar to Cartesian coordinates.
A limit cycle is a closed trajectory. Isolated means that neighboring trajectories are not closed.
A limit cycle is said to be unstable or half stable, if all neighboring trajectories approach the lemin cycle.
These systems oscillate even in the absence of external periodic force.

Consider the following two bases for ${\displaystyle R^3}$ :
$\alpha :=\{\left[\begin{array}{c}2\\ 1\\ 3\end{array}\right],\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right],\left[\begin{array}{c}3\\ 1\\ -1\end{array}\right]\}and\beta :=\{\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}-2\\ 3\\ 1\end{array}\right],\left[\begin{array}{c}2\\ 3\\ -1\end{array}\right]\}$ If $[x{]}_{\alpha }={\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]}_{\alpha }thenfind[x{]}_{\beta }$
(that is, express x in the ${\displaystyle \beta }$ coordinates).

Give a full correct answer for given question 1- Let W be the set of all polynomials ${\displaystyle a+bt+c{t}^2\in P_2}$ such that ${\displaystyle a+b+c=0}$ Show that W is a subspace of ${\displaystyle P_2,}$ find a basis for W, and then find dim(W) 2 - Find two different bases of ${\displaystyle R^2}$ so that the coordinates of $b=\left[\begin{array}{c}5\\ 3\end{array}\right]$ are both (2,1) in the coordinate system defined by these two bases

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 $\beta \text{ coordinates into }\gamma -\text{ coordinates}$

class council determined that its profit from the upcoming homecoming dance is directly related to the ticket price for the dance. Looking at past dances, the council determined that the profit pp can be modeled by the function $p(t)=-12t^2+480t+30$, where tt represents the price of each ticket. What should be the price of a ticket to the homecoming dance to maximize the council's profit? Price

Let B and C be the following ordered bases of ${\displaystyle R^3:}$
$B=(\left[\begin{array}{c}1\\ 4\\ -\frac{4}{3}\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 8\end{array}\right],\left[\begin{array}{c}1\\ 1\\ -2\end{array}\right])$
$C=(\left[\begin{array}{c}1\\ 1\\ -2\end{array}\right],\left[\begin{array}{c}1\\ 4\\ -\frac{4}{3}\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 8\end{array}\right])$ Find the change of coordinate matrix I_{CB}

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.

Consider the linear transformation ${\displaystyle U:R^3\to R^3}$ defined by $U\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}z-y\\ z+y\\ 3z-x-y\end{array}\right)$ and the bases $ϵ=\{\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)\},\gamma =\{\left(\begin{array}{c}1-i\\ 1+i\\ 1\end{array}\right),\left(\begin{array}{c}-1\\ 1\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)\}$, Compute the four coordinate matrices ${\displaystyle {\left[U\right]}_{ϵ}^{\gamma },{\left[U\right]}_{\gamma }^{\gamma },}$

Given the elow bases for $R^2$ and the point at the specified coordinate in the standard basis as below, (40 points)

$(B1=\{(1,0),(0,1)\}$&

$B2=(1,2),(2,-1)\}$(1, 7) = $3^{\ast }(1,2)-(2,1)$
$B2=(1,1),(-1,1)(3,7=5^{\ast }(1,1)+2^{\ast }(-1,1)$
$B2=(1,2),(2,1)(0,3)=2^{\ast }(1,2)-2^{\ast }(2,1)$
$(8,10)=4^{\ast }(1,2)+2^{\ast }(2,1)$
B2 = (1, 2), (-2, 1) (0, 5) =
(1, 7) =

a. Use graph technique to find the coordinate in the second basis. (10 points) b. Show that each basis is orthogonal. (5 points) c. Determine if each basis is normal. (5 points) d. Find the transition matrix from the standard basis to the alternate basis. (15 points)

Given the full and correct answer the two bases of $B1=\{\left(\begin{array}{c}2\\ 1\end{array}\right),\left(\begin{array}{c}3\\ 2\end{array}\right)\}$
$B_2=\{\left(\begin{array}{c}3\\ 1\end{array}\right),\left(\begin{array}{c}7\\ 2\end{array}\right)\}$
find the change of basis matrix from ${\displaystyle B_1\to B_2}$ and next use this matrix to covert the coordinate vector
${\overrightarrow{v}}_{B_1}=\left(\begin{array}{c}2\\ -1\end{array}\right)$ of v to its coodirnate vector
${\overrightarrow{v}}_{B_2}$