Hi,
(1) $$ \| A v_i \|^2 = < A v_i , A v_i > = < A^T A v_i , v_i > = \lambda_i $$
(2) $$ < u_i , u_j > = \frac{ < A v_i , A v_j > } { \| A v_i \| \|A v_j \| } = \frac{ < A^T A v_i , v_j > } { \| A v_i \| \|A v_j \| } = \frac{ \lambda_i \delta_{i;j} } { \| A v_i \| \|A v_j \| } $$
Also gilt \( < u_i , u_j > = 0 \) für \( i \ne j \) und wegen (1) gilt \( < u_i , u_i > = 1 \)
Also ist \( u_i \) eine orthonormal Basis.