The equivalent polar coordinates for the given rectangular coordinates. A rectangular coordinate is given as (0, -3).

Let V be a vector space, and let ${\displaystyle T:V\to V}$ be linear. Prove that ${\displaystyle T^2=T_0}$ if and only if ${\displaystyle R\left(T\right)\subseteq N\left(T\right).}$

A linear transformation $T:{\mathbb{R}}^3\to {\mathbb{R}}^2$ whose matrix is
$\left(\begin{array}{ccc}1& 3& 3\\ 2& 6& -3.5+k\end{array}\right)$
is onto if and only if $k\ne$

The coefficient matrix for a system of linear differential equations of the form $y_1=Ay$
has the given eigenvalues and eigenspace bases. Find the general solution for the system
${\lambda }_1=3+i\Rightarrow \left\{\left[\begin{array}{c}2i\\ i\end{array}\right]\right\},{\lambda }_2=3-i\Rightarrow \left\{\left[\begin{array}{c}-2i\\ -i\end{array}\right]\right\}$

Rotate a triangle with vertices (2, 1). (4. 1). (3, 3) by 90 degrees (clock wise) and then Translate it by Tx 4 and Ty 3 and finally rotate it by 30 degrees (anti-clock wise). (cos90=0, sin90 1. cos 30 = 0.866, sin30 =0.5)
T1 is Rotation with 90 T2 is Translation T3 is Rotation with 30
What is correct order of transformation matrix to multiply the metrix?
T3 T2 T1 or T1 T2 T3?
Should i follow the order given by the question or reverse order?

Find, correct to the nearest degree, the three angles of the triangle with the given vertices. P(2, 0), Q(0, 3), R(3, 4)

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

What are polar coordinates? What equations relate polar coordi-nates to Cartesian coordinates? Why might you want to change from one coordinate system to the other?