Question: Let \(B=\left\{ 1-3{{x}^{2}}+2{{x}^{3}},x+2{{x}^{2}}-3{{x}^{3}},1-3x-8{{x}^{2}}+7{{x}^{3}},2+x-5{{x}^{2}}+5{{x}^{3}} \right\}\) be the set of vectors in \({{P}_{3}}\)

1. Is the set \(B\) a basis for \({{P}_{3}}\) ? Justify. If it is not a basis for \({{P}_{3}}\) , then extend \(B\) to a basis for \({{P}_{3}}\).
2. Use the basis found in part (a) to find the coordinate vector of \(f=-1-3x-5{{x}^{2}}+11{{x}^{3}}\).

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