I. Define the mapping L: F5 P3 (F) such that 24 =(-1+23+224 +25) + (2x1-25)t...

 

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I. Define the mapping L: F5 → P3 (F) such that 24 =(-1+23+224 +25)² + (2x1-25)t + (3x1 +3 +223 +55) where x1, x2, x3, 4, 5 EF. 1. Show that is a linear transformation (7 points). 2. Find a basis for ker Land determine its dimension (8 points). 3. Based on the value of dim ker L and the rank-nullity theorem, find rank L. Is Lis an epimorphism? Justify your answer. (6 points). 4. What is the standard basis A of the vector space 5? Find the matrix representing Lwith respect to bases A and B={t2t+1,t - 1,t + 1} of P3 (F). (8 points) I. Define the mapping L: F5 → P3 (F) such that 24 =(-1+23+224 +25)² + (2x1-25)t + (3x1 +3 +223 +55) where x1, x2, x3, 4, 5 EF. 1. Show that is a linear transformation (7 points). 2. Find a basis for ker Land determine its dimension (8 points). 3. Based on the value of dim ker L and the rank-nullity theorem, find rank L. Is Lis an epimorphism? Justify your answer. (6 points). 4. What is the standard basis A of the vector space 5? Find the matrix representing Lwith respect to bases A and B={t2t+1,t - 1,t + 1} of P3 (F). (8 points)