Standard Matrix Question

**flaming** · February 6th 2011, 01:24 PM

1. Find the standard matrix defined by the linear transformation

T(x,y,z) = (x-2y, -2x-y+z,-z)

is it going to be $\left(\begin{array}{ccc}1&-2&0\\-2&-1&1\\0&0&-1\end{array}\right)$

2. Let $T_1$ be the linear transformation corresponding to a rotation by an angle of $\theta_1$ about the x axis in $R^3$ and Let $T_2$ be the linear transformation corresponding to a rotation by an angle of $\theta_2$ about the z axis in $R^3$

will the standard matrix be

$\left(\begin{array}{ccc}1&0&0\\0&cos \theta& -sin \theta\\0&sin \theta&cos \theta\end{array}\right)$

and

$\left(\begin{array}{ccc}cos \theta&-sin \theta&0\\sin \theta&cost \theta& 0\\0&0&1\end{array}\right)$

Am I right with the above answers?

**Ackbeet** · February 7th 2011, 02:15 AM

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 $\theta_{1}$ and $\theta_{2},$ so that's what you'd better have.
2. It's considered better practice to use non-italicized function names: $\cos(\theta)$ is better than $cos(\theta).$
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: $\sin(\theta),$ etc.

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