Use the Gram-Schmidt orthonormalization process to transform the given basis for R? into an orthonormal basis. Use the dot product on R?and use the vectors in the order in which they are given. B = {(1,2) , (–1,0)} o{1, 2), (-승, 음)} o {(* 2 2 O {(-1,0). (0, 1)} O {(-1,0) , (0, 2)}

Find the determinant of the vectors (2,4,0),(0,4,6),(8,0,6) with respect to the base (2,0,0),(0,2,0),(0,0,2).

Determine if the vector product <A,B> = a0b0 defines a valid inner product on P2(R) for any A=a0+a1x+a2x^2 and B=b0+b1X+b2x2. Explain your answer clearly.

Explain how fixing a basis v1, · · , Vn of V associates to a quadratic form a corresponding function in n variables of the form q(x1,· . for a certain symmetric matrix A = (aj). , Xn) = Li=1 aijXiXj, .