Exercise 1.6.1. Let V be the space of real polynomials of degree less than n. So dim V = n. Then for each a ER, the eval

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Exercise 1 6 1 Let V Be The Space Of Real Polynomials Of Degree Less Than N So Dim V N Then For Each A Er The Eval 1 (68.33 KiB) Viewed 42 times

Exercise 1.6.1. Let V be the space of real polynomials of degree less than n. So dim V = n. Then for each a ER, the evaluation ev, is a dual vector. [p(a) For any real numbers 01, ..., Q ER, consider the map L: V + R" such that L(P) = [plan)] 1. Write out the matric for L under the basis 1, 1,...,2"- for V and the standard basis for R". (Do you know the name for this matric?) 2. Prove that L is invertible if and only if a1,..., or are distinct. (If you can name the matriz L, then you may use its determinant formula without proof) 3. Show that eval,..., even form a basis for V* if and only if all a1,..., or are distinct. 4. Set n = 3. Find polynomials P-1, Po,P1 such that p:6) = dij for i, j e{-1,0,1}. 5. Set n = 4, and consider ev-2, ev-1, evo, ev1, ev, E V*. Since dim V* = 4, these must be linearly dependent. Find a non-trivial linear combination of these which is zero.