Mbzuai Entry Exam Sample Questions -

Given vectors ( u = (1, 2, -1) ), ( v = (0, 1, 3) ). Compute the projection of ( u ) onto ( v ).

def func(x, lst=[]): lst.append(x) return lst print(func(1)) print(func(2)) Write a Python function normalize(X) that takes a 2D numpy array X (samples × features) and returns a zero-mean, unit-variance normalized version (feature-wise). Do not use sklearn . mbzuai entry exam sample questions

If ( A ) and ( B ) are square invertible matrices, then ( (A + B)^-1 = A^-1 + B^-1 ). Explain briefly. Section 2: Calculus & Optimization (25%) Question 4: Find the gradient ( \nabla f(x,y) ) of ( f(x,y) = \ln(1 + e^xy) ). Then compute the directional derivative at ( (1,0) ) in the direction of ( (1,1) ). Given vectors ( u = (1, 2, -1) ), ( v = (0, 1, 3) )

Consider ( f(x) = x^3 - 2x + 1 ). Perform one iteration of gradient descent starting at ( x_0 = 1 ) with learning rate ( \eta = 0.1 ). What is ( x_1 )? Do not use sklearn