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.

Let us consider a vector space $E$ of dimension $n+1$ and a line $L\subset E$, which corresponds to the point $x\in \mathbb {P(E)}$. The tangent space $T_x\mathbb P (E)$ is canonically isomorphic to the space of linear maps $\mathcal{L}(L,E/L)$ [Harris,Algebraic Geometry, page 200, where it is even done for Grassmannians].

Hence we get a canonical isomorphism $T_x\mathbb P (E)=L^\ast \otimes E/L$ , transformed into $T_x \mathbb P(E)\otimes L=E/L$

which we write as an exact sequence $0 \to L \to E \to T_x \mathbb P E \otimes L \to 0$

This was just over the point $x\in\mathbb P(E)$. If we globalize this over the whole of $\mathbb P(E)$ we get the exact sequence of locally free sheaves [Recall that the fibre at x of the tautological line bundle $\mathcal O (-1)$ is precisely $L$]

$0\to \mathcal O(-1) \to \mathcal O ^{n+1} \to T \mathbb P (E)\otimes \mathcal O (-1) \to 0$

By tensoring this exact sequence by the invertible sheaf $\mathcal O (1)$ we obtain the euler sequence.