Prof. X has two brilliant students Y and Z inhis 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 asksY and Z to find a basis. Y
produces a set S with n elements. But Zbeing 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 asksthe class if the set S
produced by Y could be a basis without telling them whatit is. While student
U says yes, student W says need not and Prof. Xsays that both U and W
could be correct. Justify the statement of Prof. X with suitableexamples of V
over F, n, S and T.
