Do Gram-Schmidt orthogonalization for the vectors in P₂(R) f₁ = 1, f2 = √√3x, f3= x² + x + 1 The inner Product in P2 (R) is consider by and then orthonormalize it. = 19 (² g(x)h(x) dx

4. Consider the function f(x) = (a¹x)(b¹x) where a, b, and x are n-dimensional vectors. a. Find Vf(x). b. Find the Hessian F(x).

2.2 please solve it on paper

The quadratic form is positive-semidefinite but not positive definite negative-semidefinite but not negative definite indefinite qA (x, y) = x² + 2xy + 3y² negative definite positive definite

i) Find all values of z such that 25 = -1+ v3i. Plot your solutions on a complex plane. ii) For a complex function f(2) = u(x, y) + iv(x, y) with u and v real, the Cauchy-Riemann conditions state that f(2) is analytic if and only if dv du and ду du dx dy Let fi(2) and f2(2) be two analytic functions: prove that the function g(2) defined by g(2) = fi(2)f2(2) is also an analytic function. iii) Using the result of part (ii), or otherwise, determine whether the function f(z) = ze is analytic.