Basis and linear dependence III
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Exercise:
Let V be a vector space over K and suppose that V Spv_..v_n for some v_..v_nin V. Let u_...u_min V be linearly indepent. Then mleq n and moreover one can delete m vectors from the list v_...v_n s.t. the remaining list say v_'...v_n-m' together with u_..u_m still span V i.e. Spu_...u_mv_'...v_n-m'V.
Solution:
Proof of the textcolorbluemoreover-part rest can be found in Lemma D. In the proof of Lemma D we have already seen that mleq n. Now apply the procedure described in the proof of Lemma D for steps j...m. After m steps we arrive at a list u_..u_mv_'...v_n-m' of the type claimed by Lemma D'.
Let V be a vector space over K and suppose that V Spv_..v_n for some v_..v_nin V. Let u_...u_min V be linearly indepent. Then mleq n and moreover one can delete m vectors from the list v_...v_n s.t. the remaining list say v_'...v_n-m' together with u_..u_m still span V i.e. Spu_...u_mv_'...v_n-m'V.
Solution:
Proof of the textcolorbluemoreover-part rest can be found in Lemma D. In the proof of Lemma D we have already seen that mleq n. Now apply the procedure described in the proof of Lemma D for steps j...m. After m steps we arrive at a list u_..u_mv_'...v_n-m' of the type claimed by Lemma D'.