Exercise:
Let Vlangle rangle be an inner product space over KmathbbR or mathbbC and let ntextdimV. Let v_...v_n be a basis of V. Define a new list of vectors w_...w_n by induction: w_:v_w_jv_j-_i^j-fraclangle v_jw_i ranglelangle w_iw_i rangle for j...n. Then abcliste abc w_...w_n is an orthogonal basis for V. abc leftfracw_||w_||...fracw_n||w_n||right is an orthonormal basis for V. abc forall leq jleq n we have textSpv_...v_jtextSpw_...w_jtextSpleftfracw_||w_||...fracw_n||w_n||right. abcliste

Solution:
Proof. We claim that forall leq jleq n w_j is well defined i.e. w_...w_j-neq and moreover w_...w_j is an orthogonal system with textSpw_..w_jtextSpv_...v_j. We prove this by induction on j. For j w_v_ so the claim is obvious v_neq because it is a part of a basis. Let leq jleq n. Ase the claim holds for w_...w_j-. Recall that w_jv_j-_i^j-fraclangle v_jw_i ranglelangle w_iw_i rangle w_i. Note that w_j is well defined because w_ineq forall leq ileq j-. We claim that w_jneq . Indeed if w_j then v_ji^j-fraclangle v_jw_i ranglelangle w_iw_i rangle w_i Longrightarrow v_jin textSpw_..w_j-textSpv_...v_j- which is impossible because v_...v_j are linearly indepent. This shows w_jneq . By the induction hypothesis w_...w_j- is an orthogonal system. So we have to prove that w_jperp w_...w_j-. Indeed let leq kleq j- then langle w_jw_k ranglelangle v_jw_k rangle+_i^j-fraclangle v_jw_i ranglelangle w_iw_i rangle langle w_iw_krangle langle v_jw_kranglfraclangle v_jw_k ranglelangle w_kw_k rangle langle w_kw_krangle. This shows w_...w_j is an orthogonal system. To complete the induction it remains to show that textSpw_...w_jtextSpv_...v_j. By the definition of w_j we have w_jin textSpw_...w_j-v_jtextSpv_...v_j-v_j. Longrightarrow textSpw_...w_j-w_jsubseteq textSpv_...v_j-v_j. But both these vector spaces have dim j because v_...v_j are linearly indepent and also w_...w_j are linearly indepent being an orthogonal system. Longrightarrow textSpw_...w_j-w_jtextSpv_...v_j-v_j.