## Question

###### Consider the basis S = {1,1,22 in the vector space Pz, the set of all polynomials of degree at most 2, where Pz is a subspace of L?[0,1]. (a) Show that S = {1,2,22} is not an orthogonal set in L?[0,1]. (b) Use the Gram-Schmidt algorithm to generate an orthonormal basis E = {e1,e2, e3_ from S Use E to find the polynomial p = p(z) of degree at most 2 that best approximates 1/2 < x < 1 f() = 0 0 < r < 1/2 (i.e , find the (orthogonal) projection of f onto the space Pz spanned by E)_ (d S

Consider the basis S = {1,1,22 in the vector space Pz, the set of all polynomials of degree at most 2, where Pz is a subspace of L?[0,1]. (a) Show that S = {1,2,22} is not an orthogonal set in L?[0,1]. (b) Use the Gram-Schmidt algorithm to generate an orthonormal basis E = {e1,e2, e3_ from S Use E to find the polynomial p = p(z) of degree at most 2 that best approximates 1/2 < x < 1 f() = 0 0 < r < 1/2 (i.e , find the (orthogonal) projection of f onto the space Pz spanned by E)_ (d Sketch p and f on the same set of axes.

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