2. In the vector space II3 [0, 1] of all polynomials on the unit interval [0, 1] of degree not greater than 3, the following polynomials are defined, = x + 3x² + 3x³, p2(x) = x(1 − x), p3(x) = cx + x² – 3x³, P₁(x): where c is a given constant. (a) Find all values of c E R for which {P1, P2, P3} are linearly dependent. (b) The set B = {1, P₁, P2, P3} is a basis for II3 [0, 1] for those values of c for which {P1, P2, P3} are linearly independent. Find the coordinates [4] of the polynomial q(x) = (5x³ 3x)/2 in the basis B.

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A: In this problem ,we use the concepts of linear combination.

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A: DETAILED SOLUTION IS PROVIDED BELOWExplanation:

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