Here, we have sown vectors and matrixes by bold phase, elements and variables by lower case letters. Let a function f: R R be defined as: f(x) = [(* ,x2)]. Then the Jacobian of f is defined as following matrix: [afaf] Then aft Or in component wise notation ij axj Now, let composition function g to be defined as: Q-1. Let W = + ag(x) ax Q-3. If z = x, then what is af ????1 = ???? af ???? af1 x af2 = ????2 af Lx ????z g(x) = [9 (f(x), 1(x))] |1 |g2 x = (ag f Or, = **-**-*)( af4 Our purpose in this problem is to use these definitions to derive a simple form of gradient descent formulations in Lecture 24. You may answer the following questions either in component wise or matrix and vector wise. [W11 W12] W21 W22 ] and x = [X2] and z = Wx. Then find (the Jacobian) Q-2. Using Q-1, if z = x" W, then what is ???? + + , f. Ofz'????x agz afz af ax N Q-4. If z = f(x) and x = = [X], such that z = f(x) then what is x wise)? Q-5. Now let define a loss function to be J(z) = ||zy||2, where z = | Then find z awy or z x z Q-6. In order to complete our formulations, we need However, this is a complicated task. Indeed, in our case, this is a 2 x 2 x 2 tensor. But, in our calculations we need which is awy a vector. So, having z = Wx find- z for general cases of k, i and j. Then through this write awuj down the vector- Note that in this derivation we did not have the hidden layer. (component = [2] and y = [] aj z Q-7. Now, if we let in Q-5 to be 8, then write down in terms of 8 and - Finally, awy aj write down