Integral over tetrahedron Let $S$ be the getrahedron in $\mathbb{R^3}$ having vertices $(0,0,0),(1,2,3),(0,1,2),(-1,1,1)$. Evaluate $\int_{S}f$ where $f(x,y,z) = x+2y-z$. You may use a suitable linear transformation as a coordinate change

How do I solve this? As I've never studied mathematics before jumping into duplicate degree, I have very weak background on mathematics. Anybody help me? 
Thanks! 
 A: OK, here's the structure of the calculation
Step 1: find a linear map which maps the tetrahedron to something easy to parametrize.
Clearly a linear map can only take 0 to 0, so we can only play with the three other vectors.
We use the linear map defined by $(1,2,3)\mapsto(1,0,0)$, $(0,1,2)\mapsto(0,1,0)$ and $(-1,1,1)\mapsto(0,0,1)$.
Since these three vectors are independent (why?) this is a well defined linear mapping.
Moreover, since we map to a standard base we know that its inverse will be given by the matrix  $\left(\begin{matrix}1 & 0 & -1\\
2 & 1 & 1\\
3 & 2 & 1
\end{matrix}\right)$.
We denote this map $T$.
(If you are unsure about something we did so far you should revise linear algebra).
Step 2: Find a parametrization
Geomerically, the new tetrahedron is much easier to picture.
I'll leave the details to you, but it is parametrized by $\{(x,y,z) | 0\le x\le 1, 0\le y \le 1-x, 0\le z \le 1-(x+y)\}$
Note that the boundaries on $z$ are a function of $x,y$ and the boundaries on $y$ is a function of $x$ which will be useful for integration.
Step 3: use variable substitution.
Let $V$ denote the tetrahedron, then
$$\int_V f dv = \int_{T(V)} f\circ T |DT| dv$$
Note that $DT$ is simply the matrix representing $T$ (why?) which is the inverse of the matrix from before.
I'll leave you again with the details of calculating the matrix and its determinant, but denote for now $|DT|=a$. I also leave to you to calculate $T\circ f$, which we will denote for now as $g$.
Putting everything in place we get that
$$\int_{T(V)} f\circ T |DT| dv = a\int_{T(V)} g dv$$
and using the parametrization from before
$$\int_{T(V)} g dv = \int_0^1 \int_0^{1-x} \int_0^{1-x-y} g(x,y,z) dzdydx$$
which should turn out an elementary calculation
