MSCA 37016 Assignment 2

#Question 1 #a. 3. Because the left part of second equation equals to # the two times the left part of the first equation, any number other than # 2*1 will give no solution, #b. 2. For the same reason as in part a, if q = 2, two equations will be the same. # Two variables and one equation give infinte solution.

#Question 2 #[[2,3,1|12],[-2,3,-2 | 1],[1,-1,4 |16]] #2*L3 + L2 => L3 = [0,1,6 | 33] #L2 + L1 => L2 = [0,6,-1 | 13] #L2 <=> L3 #L3 -6*L2 => L3 = [0,0,-37 | -185] #z = -185/(-37) = 5 #y+5*6 = 33 => y =3 #2*x +3*3+1*5 = 12 => x = -1 #x = -1, y = 3, z = 5

#Code for question 2 import numpy as np a = np.array([[2,3,1],[-2,3,-2],[1,-1,4]]) b = np.array([12,1,16]) x = np.linalg.solve(a, b) x

#Question 3 #Rank is the number of pivot columns #a. L4 - L3 => L4 = [0,0,0,0,-3], so the rank(A) = 4 #b. L2 - 2*L1 => L2 = [0,0,1,-2] #L3 - L1 => L3 = [0,0,-2,5] #L3 + 2*L2 => L3 = [0,0,0,1], so the rank(A) = 3

#Code for the question 3a Aa = np.array([[1,3,1,2,0],[0,0,2,1,3],[0,0,0,3,2],[0,0,0,3,1]]) np.linalg.matrix_rank(Aa)

#Code for the question 3b Ab = np.array([[-1,1,0,-1],[-2,2,1,-4],[-1,1,2,3]]) np.linalg.matrix_rank(Ab)

#Question 4 #They are linearly dependent, because there are three vectors in R2.