Let Br = {1,..., k} be the standard k-element set. Recall that the matrix M(f) associated...

 

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Let Br = {1,..., k} be the standard k-element set. Recall that the matrix M(f) associated to a map f : Bm → Bn is the array with n rows and m columns a11 a12 aim a21 a22 azm ... M(f) = an1 an2 anm where aij = 1 precisely when f(j) = i and aij = 0 otherwise. (Side remark: we may think of M(f) as a list of binary values (ai;) indexed by Bn x Bm, or more precisely, as a map Bn x Bm → {0, 1}.) 1. Let f : B3 -+ Ba be the map defined by f(x) = -2x2+ 7x – 2. Compute M(f). 2. If r : B3 → B3 is a permutation of the domain, and l: B4 → B4 is a permutation of the codomain, then we may change the map f into the following map f' = lofor: B3 + BA. 2a) Without permuting the codomain, is it possible to choose r so that the matrix of f' is equal to 1 0 1 0 0 1 0 0 0 If so, how? 2b) On the other hand, if we keep the domain fixed and permute the codomain, is it possible to obtain the matrix above for f'? If so, how? Let Br = {1,..., k} be the standard k-element set. Recall that the matrix M(f) associated to a map f : Bm → Bn is the array with n rows and m columns a11 a12 aim a21 a22 azm ... M(f) = an1 an2 anm where aij = 1 precisely when f(j) = i and aij = 0 otherwise. (Side remark: we may think of M(f) as a list of binary values (ai;) indexed by Bn x Bm, or more precisely, as a map Bn x Bm → {0, 1}.) 1. Let f : B3 -+ Ba be the map defined by f(x) = -2x2+ 7x – 2. Compute M(f). 2. If r : B3 → B3 is a permutation of the domain, and l: B4 → B4 is a permutation of the codomain, then we may change the map f into the following map f' = lofor: B3 + BA. 2a) Without permuting the codomain, is it possible to choose r so that the matrix of f' is equal to 1 0 1 0 0 1 0 0 0 If so, how? 2b) On the other hand, if we keep the domain fixed and permute the codomain, is it possible to obtain the matrix above for f'? If so, how?

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To create the matrix of f we use the columnwise approach th... View the full answer