Dual maps and bases
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Exercise:
Let S:Vlongrightarrow W be a linear map between two finite dimensional vector spaces V and W over K. Let mathcalB be a basis for V and mathcalC be a basis for W. Consider S^*:W^*longrightarrow V^* and the bases mathcalC^* B^* for W^* V^* respectively. Then S^*_mathcalB^*^mathcalC^*S_mathcalC^mathcalB^T.
Solution:
Proof. Write mathcalBv_...v_nmathcalCw_...w_n. A:S_mathcalC^mathcalB Aa_ij. A satisfies Sv_j _i^n a_ijw_i. Now for w_j^*in mathcalC^* we have S^*w_j^*w_j^*circ S _i^n w_j^*circ Sv_i v_i^* _i^n w_j^*Sv_i v_i^* _i^n w_j^*left_k^n a_kiw_kright v_i^* _i^n a_jiv_i^* The coordinates of S^*w_j^* in the basis mathcalB^*v_^*...v_n^* are leftarrayc a_j vdots a_jn arrayright row#j in A represented as a col col # j in the matrix A^T.
Let S:Vlongrightarrow W be a linear map between two finite dimensional vector spaces V and W over K. Let mathcalB be a basis for V and mathcalC be a basis for W. Consider S^*:W^*longrightarrow V^* and the bases mathcalC^* B^* for W^* V^* respectively. Then S^*_mathcalB^*^mathcalC^*S_mathcalC^mathcalB^T.
Solution:
Proof. Write mathcalBv_...v_nmathcalCw_...w_n. A:S_mathcalC^mathcalB Aa_ij. A satisfies Sv_j _i^n a_ijw_i. Now for w_j^*in mathcalC^* we have S^*w_j^*w_j^*circ S _i^n w_j^*circ Sv_i v_i^* _i^n w_j^*Sv_i v_i^* _i^n w_j^*left_k^n a_kiw_kright v_i^* _i^n a_jiv_i^* The coordinates of S^*w_j^* in the basis mathcalB^*v_^*...v_n^* are leftarrayc a_j vdots a_jn arrayright row#j in A represented as a col col # j in the matrix A^T.