The equivalent rectangular coordinates for the given polar coordinates: [left(6, -135^{circ} right)]

Assume that T is a linear transformation. Find the standard matrix of T.
${\displaystyle T={\mathbb{R}}^2\to {\mathbb{R}}^{4}suchthatT\left(e_1\right)=(7,1,7,1),}$.

$\text{and}T\left(e_2\right)=(-8,5,0,0),where{e}_1=(1,0),$

$\text{and}{e}_2=(0,1)$

Consider the bases $B=\left(\begin{array}{c}\left[\begin{array}{c}2\\ 3\end{array}\right],\left[\begin{array}{c}3\\ 5\end{array}\right]\end{array}\right)of{R}^{2}andC=(\begin{array}{c}\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]\end{array},\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right])of{R}^3$.
and the linear maps $S\in L(R^2,R^3)andT\in L(R^3,R^2)$ given given (with respect to the standard bases) by $[S{]}_{E,E}=\left[\begin{array}{cc}2& -1\\ 5& -3\\ -3& 2\end{array}\right]and[T{]}_{E,E}=\left[\begin{array}{ccc}1& -1& 1\\ 1& 1& -1\end{array}\right]$ Find each of the following coordinate representations. ${\displaystyle \left(b\right){\left[S\right]}_{E,C}}$
${\displaystyle \left(c\right){\left[S\right]}_{B,C}}$

Consider the following system.
$\{\begin{array}{l}y=30-x\\ 2y=-x^2+16x-12\end{array}$

Show that if u is a vector in R^{2} or R&3, then ${\displaystyle u+(-1)u=0}$
${\displaystyle u+(-1)u=0}$

To find: The equivalent polar equation for the given rectangular-coordinate equation.
Given:
${\displaystyle x=r\cos \theta }$
${\displaystyle y=r\sin \theta }$
b. From rectangular to polar:
${\displaystyle r=\pm \sqrt{x^2+y^2}}$
${\displaystyle \cos \theta =\frac{x}{r},\sin \theta =\frac{y}{r},\tan \theta =\frac{x}{y}}$
Calculation:
Given: equation in rectangular-coordinate is ${\displaystyle y=x}$.
Converting into equivalent polar equation -
${\displaystyle y=x}$
Put ${\displaystyle x=r\cos \theta ,y=r\sin \theta ,}$
${\displaystyle \Rightarrow r\sin \theta =r\cos \theta }$
${\displaystyle \Rightarrow \frac{\sin \theta }{\cos \theta }=1}$
${\displaystyle \Rightarrow \tan \theta =1}$
Thus, desired equivalent polar equation would be ${\displaystyle \theta =1}$

Discuss: Different Coordinate Systems As was noted in the overview of the chapter, certain curves are more naturally described in one coordinate system than in another. In each of the following situations, which coordinate system would be appropriate: rectangular or polar? Give reasons to support your answer.
a) You need to give directions to your house to a taxi driver.
b) You need to give directions to your house to a homing pigeon.

We can describe the location of a point in the plane using different ________ systems. The point P shown in the figure has rectangular coordinates ( _ , _ ) and polar coordinates ( _, _).