find scalars c1, c2, and c3 for which the equation is satisfied. 19. c1(1, −1, 0) + c2(3, 2, 1) + c3(0, 1, 4) = (−1, 1, 19)

= 12. (a) Find the unique quadratic polynomial y ax²+bx+c that passes through the three points (1, 1), (3,5), and (-2, 0) in the x-y plane. (Hint: this will involve a linear system where a, b, and c are the unknowns.) (b) How many data points (x, y) should be needed to uniquely determine the coefficients of an nth degree polynomial anx" +an-1² -1xn-1+...+ a₁x+ao? Explain your answer in terms of linear systems.

Find an orthogonal change of variables, x = Py, that transforms the quadratic form q(x) = x² + x²+2x1x2 − 2x2x3 into a quadratic form no cross-product form. Write the new quadratic form.

Express w as a linear combination of u and v. 1 (-) 8 u 1 5 w=u+v (Type integers or fractions.) W - 3 0