A constant level contour of a continuous cost function VR R, corresponding to a given cost value CER is the set: Ve{ER" | V (x) = c} (a) Given points ro EV and rER" suppose that the vector point to which we write as (10) V. Provide a necessary perpendicularity. Does the function V need to be differentiable 1-0 is perpendicular and sufficient condition at ? Please, explain. (1) to V at the for such (b) Prove that, whenever it exists, the gradient of V at 20 V, denoted by VV (20), is a vector perpendicular to V at ro. Prove that it points towards increasing values of V. (c) Suppose that sER" is a descent direction of a cost function VRR at a given point r E R in the sense that VV (r)s 0 that guarantees V (r) > V (x + ws). (2) Q.2 (a) Prove that any symmetric positive definite matrix is non-singular. Do not use the eigenvalues in your argument. (b) Prove that all the eigenvalues of a symmetric, real matrix, are real numbers. A real matrix is one whose all entries are real numbers. (c) Provide a definition of a rank of a matrix. Define possible matrix transformations matrix rank. Prove your assertions. that preserve Q.3 / (a) Let A Rmx be a nonzero matrix Determine whether any global solution to the least squares problem arg min{||Az yall | R") (3) corresponds to an orthogonal projection of y ER onto the range of A. (b) Suppose that A is a matrix a unique solution for any such Ar = 0 for any r = 0. Does it mean that the equation Ar = y has y? Explain your answer in full. (c) Suppose that A is an n x m matrix full of zeros. Derive the More-Penrose pseudo-inverse of A. Explain in detail how you obtain the result