Let $L=\mathbf{P}^m\subset\mathbf{P}^n$ be a linear subspace with $1\leq m\leq n-2$, say defined by $x_{m+1}=\ldots=x_n=0$. Let the blow up $X$ of $\mathbf{P}^n$ with center $L$. The exceptional divisor is of the form $E=\mathbf{P}^m\times\mathbf{P}^{n-m-1}$. Then the blow up morphism contracts second component. Alternatively,by Fujiki-Nakano criterion (Remark 11.10 in Fourier Mukai by Huybrechts), there is a morphism $$f:X\to Y$$ which is a blow up of $\mathbf{P}^{n-m-1}\subset Y$ with exceptional divisor $E$ and whose retraction to $E$ is the contraction of the first component $$\pi':\mathbf{P}^m\times\mathbf{P}^{n-m-1}\to \mathbf{P}^{n-m-1}$$
My question is: Is $f$ the blow up of $\mathbf{P}^n$ with center $x_0=\ldots=x_m=0$? If not, what does $Y$ look like?
The second contraction is not birational. In fact, it is a $\mathbb{P}^{m+1}$-bundle over $\mathbb{P}^{n-m-1}$ induced by the linear projection $\mathbb{P}^n \dashrightarrow \mathbb{P}^{n-m+1}$ out of $\mathbb{P}^m$.