
Standard Matrix Question
1. Find the standard matrix defined by the linear transformation
T(x,y,z) = (x2y, 2xy+z,z)
is it going to be $\displaystyle \left(\begin{array}{ccc}1&2&0\\2&1&1\\0&0&1\end{array}\right)$
2. Let $\displaystyle T_1$ be the linear transformation corresponding to a rotation by an angle of $\displaystyle \theta_1$ about the x axis in $\displaystyle R^3$ and Let $\displaystyle T_2$ be the linear transformation corresponding to a rotation by an angle of $\displaystyle \theta_2$ about the z axis in $\displaystyle R^3$
will the standard matrix be
$\displaystyle \left(\begin{array}{ccc}1&0&0\\0&cos \theta& sin \theta\\0&sin \theta&cos \theta\end{array}\right)$
and
$\displaystyle \left(\begin{array}{ccc}cos \theta&sin \theta&0\\sin \theta&cost \theta& 0\\0&0&1\end{array}\right)$
Am I right with the above answers?

Your answers are close. For the rotations, I have several comments:
1. Most important: your angles need to be subscripted. That is, the problem statement has $\displaystyle \theta_{1}$ and $\displaystyle \theta_{2},$ so that's what you'd better have.
2. It's considered better practice to use nonitalicized function names: $\displaystyle \cos(\theta)$ is better than $\displaystyle cos(\theta).$
3. I would highly recommend ALWAYS putting parentheses around function arguments. I've been burned on that one before, and so rather than write in a way that people can understand, I would rather try to write in a way that no one can misunderstand. So I'd recommend writing trig functions this way: $\displaystyle \sin(\theta),$ etc.