Let us consider a vector space E of dimension n+1 and a line ![L\subset E](http://latex.mathoverflow.net/png?L%5Csubset%20E), which corresponds to the point 
![x\in \mathbb {P(E)}](http://latex.mathoverflow.net/png?x%5Cin%20%5Cmathbb%20%7BP%28E%29%7D).
The tangent space ![T\sb x\mathbb P (E)](http://latex.mathoverflow.net/png?T%5Fx%5Cmathbb%20P%20%28E%29)  is canonically isomorphic to  the space of linear maps ![\mathcal{L}(L,E/L)](http://latex.mathoverflow.net/png?%5Cmathcal%7BL%7D%28L%2CE%2FL%29) 
[Harris,Algebraic Geometry, page 200, where it is even done  for Grassmannians].

Hence we get a canonical isomorphism ![T\sb x\mathbb P (E)=L^\ast  \otimes E/L](http://latex.mathoverflow.net/png?T%5Fx%5Cmathbb%20P%20%28E%29%3DL%5E%2A%20%5Cotimes%20E%2FL) , transformed into
![T\sb x \mathbb P(E)\otimes L=E/L](http://latex.mathoverflow.net/png?T%5Fx%20%5Cmathbb%20P%28E%29%5Cotimes%20L%3DE%2FL)

 
 which we write as  an exact sequence ![0 \to L \to E \to T\sb x \mathbb P E \otimes L \to 0](http://latex.mathoverflow.net/png?0%20%5Cto%20L%20%5Cto%20E%20%5Cto%20T%5Fx%20%5Cmathbb%20P%20E%20%5Cotimes%20L%20%5Cto%200)

This was just over the point ![x\in\mathbb P(E)](http://latex.mathoverflow.net/png?x%5Cin%5Cmathbb%20P%28E%29). If we globalize this over the whole of ![\mathbb P(E)](http://latex.mathoverflow.net/png?%5Cmathbb%20P%28E%29) we get the exact sequence of locally free sheaves [Recall that the fibre at x of the tautological line bundle ![\mathcal O (-1)](http://latex.mathoverflow.net/png?%5Cmathcal%20O%20%28%2D1%29)   is precisely L]

![0\to \mathcal O(-1) \to \mathcal O ^{n+1} \to T \mathbb P (E)\otimes \mathcal O (-1) \to 0](http://latex.mathoverflow.net/png?0%5Cto%20%5Cmathcal%20O%28%2D1%29%20%5Cto%20%5Cmathcal%20O%20%5E%7Bn%2B1%7D%20%5Cto%20T%20%5Cmathbb%20P%20%28E%29%5Cotimes%20%5Cmathcal%20O%20%28%2D1%29%20%5Cto%200)

By tensoring this exact sequence by the invertible sheaf ![\mathcal O (1)](http://latex.mathoverflow.net/png?%5Cmathcal%20O%20%281%29)  we obtain the euler sequence.