Check that {1,(x+1),(x+1)^2} is a basis of the vector space of polynomial over R of degree at most 2. Find the coordinate of 3+x+2x^2 with respect to the basis.
Wronskian of the functions is
,
so the functions are independent and hence form the basis.
To find a coordinate of 3+x+2x^2 find such values of A, B, and C such that
therefore C = 2;
(B + 2 C) = 1and B = 1- 2C = -3;
(A + B + C) = 3 and A = 3 - B - C = 4.
So the coordinates are {4, -3, 2}
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