Relation dual basis and normal basis
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Exercise:
Let V be a finite dimensional vector space over K. Let mathcalBC be two bases for V Longrightarrow mathcalB^*C^* are two bases for V^*. Show the following identity: id_V^*_mathcalB^*^mathcalC^*leftid_V_mathcalC^mathcalBright^Tleftleftid_V_mathcalB^mathcalCright^Tright^-
Solution:
Proof. Let mathcalBv_...v_n mathcalCw_...w_n. A:id_V_mathcalC^mathcalB pmatrix vdots & & vdots v__mathcalC & hdots & v_n_mathcalC vdots & & vdots pmatrix. Write also Aa_ij. We have v_i_k^n a_kiw_k by definition. Let mathcalC^*w_^*...w_n^* be the dual basis of mathcalC. We apply the previous Lemma to lw_j^* and the basis mathcalB: w_j^*_i^n w_j^*v_i v_i^*_i^n w_j^*left_k^n a_kiw_kright v_i^* _i^n a_jiv_i^* Longrightarrow id_V^*_mathcalB^*^mathcalC^*A^Tleftid_V_mathcalC^mathcalBright ^T only kj remains from the inner
Let V be a finite dimensional vector space over K. Let mathcalBC be two bases for V Longrightarrow mathcalB^*C^* are two bases for V^*. Show the following identity: id_V^*_mathcalB^*^mathcalC^*leftid_V_mathcalC^mathcalBright^Tleftleftid_V_mathcalB^mathcalCright^Tright^-
Solution:
Proof. Let mathcalBv_...v_n mathcalCw_...w_n. A:id_V_mathcalC^mathcalB pmatrix vdots & & vdots v__mathcalC & hdots & v_n_mathcalC vdots & & vdots pmatrix. Write also Aa_ij. We have v_i_k^n a_kiw_k by definition. Let mathcalC^*w_^*...w_n^* be the dual basis of mathcalC. We apply the previous Lemma to lw_j^* and the basis mathcalB: w_j^*_i^n w_j^*v_i v_i^*_i^n w_j^*left_k^n a_kiw_kright v_i^* _i^n a_jiv_i^* Longrightarrow id_V^*_mathcalB^*^mathcalC^*A^Tleftid_V_mathcalC^mathcalBright ^T only kj remains from the inner