In the following problems check to see if the set S is a subspace of the corresponding vector space. If it is not, explain why not. If it is, then find a basis and the dimension. 3 (b) X1 X2 S = x3 x4 2x1 + x2 x3; x1 + 2x2 + x3 = 0 CR4 (c) (d) S = -{[]} S = a x2 0 -a , x1 x2 ER} CR² Va ER CR2×2 = {f(x), df = sin(wf), Vw #0 € R 0ER CR S = dx ER} <

For those that you did not select as an element of P_2(R). Briefly explain why it was not an element of P_2(R).

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What is a linear subspace?

Why do the set of all polynomials of degree n or less form a vector space, while the set of all polynomials of degree n or more do not?

Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned by 9x - 8x2– 4, 7x – 10x2-7 and 2x - x2. a The dimension of the subspace H is b. Is (9x- 8x2 -4,7x– 10x²– 7,2x – x²) a basis for P2? choose + Be sure you can explain and justify your answer. c. A basis for the subspace H is { }. Enter a polynomial or a comma separated list of polynomials.