Let X = {r1,..., xn} be a finite set, and Fa field. The vector space over...

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Let X = {r1,..., xn} be a finite set, and Fa field. The vector space over F generated by X is, by definition, the set consisting of "formal linear combinations" of elements of X, namely, expressions of the form a1x1 +... + antn where a; e F. If a summand in the expression above has zero coefficient, it coincides with the zero vector and we typically omit it from the expression. Similarly, we typically omit the coefficient if the summand is of the form 1x;, writing simply x; instead. Vector addition is defined componentwise, meaning that Σ (a; + b;)xi, and scalar multiplication is defined similarly, i.e. Denote this vector space by F(X). Finally, choose a labeling l : Bn X, giving rise to a basis (x(1), ..., x(n)) of F(X). Then we can promote any map f : X → X into a linear map pf : F(X) → F(X) by specifying its value on the basis: Pf(@1) = f(xu), i = 1, 2,.... 1. If f and g are maps from X to X, they may be composed; prove that the linear map associated to the composition of f and g coincides with the composition of linear maps associated to f and to g: Pfog = Pf O Pg 2. Let X be a set of cardinality 4. Fix a labeling for X and let B be the associated basis for F(X). List all matrices alerle where f ranges over all possible bijections from X to X. 3. Prove that the set of matrices found in part 2 form a group under matrix multiplication. Let X = {r1,..., xn} be a finite set, and Fa field. The vector space over F generated by X is, by definition, the set consisting of "formal linear combinations" of elements of X, namely, expressions of the form a1x1 +... + antn where a; e F. If a summand in the expression above has zero coefficient, it coincides with the zero vector and we typically omit it from the expression. Similarly, we typically omit the coefficient if the summand is of the form 1x;, writing simply x; instead. Vector addition is defined componentwise, meaning that Σ (a; + b;)xi, and scalar multiplication is defined similarly, i.e. Denote this vector space by F(X). Finally, choose a labeling l : Bn X, giving rise to a basis (x(1), ..., x(n)) of F(X). Then we can promote any map f : X → X into a linear map pf : F(X) → F(X) by specifying its value on the basis: Pf(@1) = f(xu), i = 1, 2,.... 1. If f and g are maps from X to X, they may be composed; prove that the linear map associated to the composition of f and g coincides with the composition of linear maps associated to f and to g: Pfog = Pf O Pg 2. Let X be a set of cardinality 4. Fix a labeling for X and let B be the associated basis for F(X). List all matrices alerle where f ranges over all possible bijections from X to X. 3. Prove that the set of matrices found in part 2 form a group under matrix multiplication.