The blow up, $\widetilde{X}$ of an affine variety $X \subset \mathbb A^n$ at $f_1,...,f_m$ is the closure of $\Gamma_f = \{(x,f(x)) : x \in U \}$ in $X \times \mathbb P^{m-1}$ (where $U = X \backslash Z(f_1,...,f_m)$, $f(x) := (f_1(x),...,f_m(x))$, and assuming $U \neq \emptyset$).
But I have also seen a definition something like $\widetilde{X} = \{ (x,y) \in X \times P^{m-1} : x_i y_j = x_jy_i \}$ in the case where $f_i = x_i$, (that is, when we are blowing up at the origin).
There is an easy inclusion (the first into the second), but I've been trying to figure out if these two definitions agree in general. And if they do, can anyone provide a reference? (I'm pretty hazy on schemes so if it can be argued without them it would be appreciated).
You will get the same thing so long as $$(f_1,\ldots, f_m) = (x_1, \ldots, x_n)$$ as ideals. To see this, first note that your second definition is the same as the first in the case where your functions are the $x_i$ (which it appears you already have), and then use the fact that the $f_i$ generate the same ideal to write the $x_i$ as $k[x_1, \ldots, x_n]$-linear combinations of the $f_j$. This should give you your map in the other direction.