Alternating n-linear function
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Exercise:
Let ngeq and D an n--linear function which is alternating. Let leq jleq n. Define a function E_j:M_ntimes nKlongrightarrow K by E_jA:_i^n -^i+jA_ijD_ijA. Then E_j is an n-linear function and it is alternating. If D is a determinant function then so is E_j.
Solution:
Let Ain M_ntimes nK. Clearly D_ijA is indepent of the ith row of A. Since D is n--linear the map Amapsto D_ijA is linear when viewed as a function of each of its rows except row #i. As a function of row #i this function is constant and not necessarily so in general not linear. Longrightarrow The function Amapsto A_ij D_ijA is n-linear. Now linear combination of n-linear functions are n-linear hence E_jA_i^n -^i+jA_ijD_ijA is n-linear too. We'll show now that E_j is alternating. By a previous proposition it is enough to show that E_jA whenever A has two equal adjacent rows. So write A pmatrix hdots & alpha_ & hdots hdots & vdots & hdots hdots & alpha_n & hdots pmatrix and ase that alpha_kalpha_k+ for some leq kleq n-. Let leq ileq n s.t. ineq k and ineq k+. The matrix Ai|j has two equal rows hence D_ijA because D_ijADAi|j and D is ased to be alternating. Longrightarrow E_jA-^k+jA_kjD_kjA+-^k++jA_k+jD_k+jA. textBut alpha_kalpha_k+Longrightarrow A_kjA_k+jA Longrightarrow E_jA This proves that E_j is n-linear and alternating. Finally ase that D is a determinant function i.e. it has the additional property that DI_n-. Note that I_nj|jI_n- forall j. So E_jI_n_i^n -^i+jI_n_ij D_ijI_n -^j+jD_jjI_nDI_nj|jDI_n-. This shows that E_j is a determinant function.
Let ngeq and D an n--linear function which is alternating. Let leq jleq n. Define a function E_j:M_ntimes nKlongrightarrow K by E_jA:_i^n -^i+jA_ijD_ijA. Then E_j is an n-linear function and it is alternating. If D is a determinant function then so is E_j.
Solution:
Let Ain M_ntimes nK. Clearly D_ijA is indepent of the ith row of A. Since D is n--linear the map Amapsto D_ijA is linear when viewed as a function of each of its rows except row #i. As a function of row #i this function is constant and not necessarily so in general not linear. Longrightarrow The function Amapsto A_ij D_ijA is n-linear. Now linear combination of n-linear functions are n-linear hence E_jA_i^n -^i+jA_ijD_ijA is n-linear too. We'll show now that E_j is alternating. By a previous proposition it is enough to show that E_jA whenever A has two equal adjacent rows. So write A pmatrix hdots & alpha_ & hdots hdots & vdots & hdots hdots & alpha_n & hdots pmatrix and ase that alpha_kalpha_k+ for some leq kleq n-. Let leq ileq n s.t. ineq k and ineq k+. The matrix Ai|j has two equal rows hence D_ijA because D_ijADAi|j and D is ased to be alternating. Longrightarrow E_jA-^k+jA_kjD_kjA+-^k++jA_k+jD_k+jA. textBut alpha_kalpha_k+Longrightarrow A_kjA_k+jA Longrightarrow E_jA This proves that E_j is n-linear and alternating. Finally ase that D is a determinant function i.e. it has the additional property that DI_n-. Note that I_nj|jI_n- forall j. So E_jI_n_i^n -^i+jI_n_ij D_ijI_n -^j+jD_jjI_nDI_nj|jDI_n-. This shows that E_j is a determinant function.