Answer to Question #141466 in Linear Algebra for suki

Question #141466
Is there a linear map T : P2(R) ->R2 such that T(1) = (2, 3), T(1+x) = (-2, 7), and T(1+x+x^2) =
(0, 9)? Justify your answer.
1
Expert's answer
2020-11-08T19:44:20-0500

Clearly 1,1+x,1+x+x21, 1+x,1+x+x^2 forms a basis of P2(R).P^2(\mathbb{R}). This follows from the fact that x=1+x1,x2=1+x+x21(1+x)x=1+x-1, x^2=1+x+x^2 -1-(1+x) and 1,x,x21,x,x^2 forms a basis. Now mapping any basis to any set of vectors gives a linear map. Hence the given map is linear. We need to extend the map by linearity on the entire space.


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