Let Pn denote the vector space of all real-valued polynomials of degree at most n. Let...

 

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Let Pn denote the vector space of all real-valued polynomials of degree at most n. Let W = {ƒ € P₁ : ƒ(−1) = 0, ƒ”(0) = 0}. (a) Show that W is a subspace of P₁. (Hint: Write f (x) = ax¹+bx³ + cr² + dx +e and give an equivalent description of W in terms of a, b, c, d, e.) (b) Find a basis for W (and prove that it is a basis). (c) What is the dimension of W? Let Pn denote the vector space of all real-valued polynomials of degree at most n. Let W = {ƒ € P₁ : ƒ(−1) = 0, ƒ”(0) = 0}. (a) Show that W is a subspace of P₁. (Hint: Write f (x) = ax¹+bx³ + cr² + dx +e and give an equivalent description of W in terms of a, b, c, d, e.) (b) Find a basis for W (and prove that it is a basis). (c) What is the dimension of W?