Q1 a) Consider the linear system given as follows:

c1x + 2y + 7z = b1

c1x + c1y + 4z = b2

c1x + c1y + c1z = b3

In this linear system we have an unknown constant named c1. Find out the three values of c1 for which this linear system will fail to have exactly three pivots.

b) Prof. X has two brilliant students Y and Z in his class. He introduces the concept of vector spaces, bases and dimension in his 5th session. As an exercise, he gives a vector space V of dimension n and asks Y and Z to find a basis. Y produces a set S with n elements. But Z being lazy, takes the set S, removes a vector and adds a new vector to it creating a new set T. Prof. X looks at set T and confirms to class that it is a basis. He then asks the class if the set S produced by Y could be a basis without telling them what it is. While student U says yes, student W says need not and Prof. X says that both U and W could be correct. Justify the statement of Prof. X with suitable examples of V over F, n, S and T