Problem 1 (10 points): You need two write two functions: inconsistentSystem(B) which takes a matrix B in echelon form, and returns True if it represents an inconsistent system, and False otherwise; and backsubstitution(B) which returns the reduced row echelon form matrix of B. Please complete the following Python codes.

import numpy as np import warnings

def swapRows(A, i, j): """ interchange two rows of A operates on A in place """ tmp = A[i].copy() A[i] = A[j] A[j] = tmp

def relError(a, b): """ compute the relative error of a and b """ with warnings.catch_warnings(): warnings.simplefilter("error") try: return np.abs(a-b)/np.max(np.abs(np.array([a, b]))) except: return 0.0

def rowReduce(A, i, j, pivot): """ reduce row j using row i with pivot pivot, in matrix A operates on A in place """ factor = A[j][pivot] / A[i][pivot] for k in range(len(A[j])): # we allow an accumulation of error 100 times larger # than a single computation # this is crude but works for computations without a large # dynamic range if relError(A[j][k], factor * A[i][k]) < 100 * np.finfo('float').resolution: A[j][k] = 0.0 else: A[j][k] = A[j][k] - factor * A[i][k]

# stage 1 (forward elimination) def forwardElimination(B): """ Return the row echelon form of B """ A = B.copy() m, n = np.shape(A) for i in range(m-1): # Let lefmostNonZeroCol be the position of the leftmost nonzero value # in row i or any row below it leftmostNonZeroRow = m leftmostNonZeroCol = n ## for each row below row i (including row i) for h in range(i,m): ## search, starting from the left, for the first nonzero for k in range(i,n): if (A[h][k] != 0.0) and (k < leftmostNonZeroCol): leftmostNonZeroRow = h leftmostNonZeroCol = k break # if there is no such position, stop if leftmostNonZeroRow == m: break # If the leftmostNonZeroCol in row i is zero, swap this row # with a row below it # to make that position nonzero. This creates a pivot in that position. if (leftmostNonZeroRow > i): swapRows(A, leftmostNonZeroRow, i) # Use row reduction operations to create zeros in all positions # below the pivot. for h in range(i+1,m): rowReduce(A, i, h, leftmostNonZeroCol) return A

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# If any operation creates a row that is all zeros except the last element, # the system is inconsistent; stop. def inconsistentSystem(A): """ B is assumed to be in echelon form; return True if it represents an inconsistent system, and False otherwise """

def backsubstitution(B): """ return the reduced row echelon form matrix of B """

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