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Exercise:
The map Phi:K^l longrightarrow textSolA ln-textrankA is linear. Moreover it is an isomorphism.
Solution:
Proof of linearity. The k-th non-zero row of A' gives the following : x_j_k&+textlinear combination of the free vars. that come after j_k x_jk_q|leq qleq l i_q j_k a_kq'x_iq Longrightarrow x_j...x_jr dep linearly on the free variables x_i...x_il. This proves the linearity of Phi. Proof of isomorphism. To prove that Phi is an isomorphism it's enough to show that textKerPhi because Phi:K^l longrightarrow textSolAl-textdim space. But this clearly hold: if lambdalambda_...lambda_lextKerPhi i.e. if Philambda_...lambda_l then by definition of Phi we have lambda_...lambda_l because each lambda_k is an entry in Philambda_...lambda_l.
The map Phi:K^l longrightarrow textSolA ln-textrankA is linear. Moreover it is an isomorphism.
Solution:
Proof of linearity. The k-th non-zero row of A' gives the following : x_j_k&+textlinear combination of the free vars. that come after j_k x_jk_q|leq qleq l i_q j_k a_kq'x_iq Longrightarrow x_j...x_jr dep linearly on the free variables x_i...x_il. This proves the linearity of Phi. Proof of isomorphism. To prove that Phi is an isomorphism it's enough to show that textKerPhi because Phi:K^l longrightarrow textSolAl-textdim space. But this clearly hold: if lambdalambda_...lambda_lextKerPhi i.e. if Philambda_...lambda_l then by definition of Phi we have lambda_...lambda_l because each lambda_k is an entry in Philambda_...lambda_l.