Let T: V V be a linear transformation on a finite dimensional F-vector space V. We...

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Let T: V→ V be a linear transformation on a finite dimensional F-vector space V. We define the determinant of T (denoted by det (T)) as follows: we choose an ordered basis 3 for V, and then define det(T) as the determinant of the matrix representation det([T]s) of T. We want to show that this gives a well-defined function det: L(V, V) → F. Let L: R2 R2 be a linear transformation defined as (2a + b, 3a - b). (2) Let y (1) Let 3' = {(1,0), (0, 1)} be the standard ordered basis for R2. Compute det([L],3¹). = L(a, b) det ([L]). = {(1, 0), (1, -1)} be another ordered basis for R2. Compute (3) Prove that the above definition for the determinant of a linear transfor- mation T: V→ V is well-defined by showing that det ([7]) = det([T],) for any ordered bases 3,7 for V (Hint: use the change of coordinate matrix). Let T: V→ V be a linear transformation on a finite dimensional F-vector space V. We define the determinant of T (denoted by det (T)) as follows: we choose an ordered basis 3 for V, and then define det(T) as the determinant of the matrix representation det([T]s) of T. We want to show that this gives a well-defined function det: L(V, V) → F. Let L: R2 R2 be a linear transformation defined as (2a + b, 3a - b). (2) Let y (1) Let 3' = {(1,0), (0, 1)} be the standard ordered basis for R2. Compute det([L],3¹). = L(a, b) det ([L]). = {(1, 0), (1, -1)} be another ordered basis for R2. Compute (3) Prove that the above definition for the determinant of a linear transfor- mation T: V→ V is well-defined by showing that det ([7]) = det([T],) for any ordered bases 3,7 for V (Hint: use the change of coordinate matrix).