Answer to Question #45758 in Engineering for sapna

Question

Which of the following statements are true and which are false? Justify your answer with a
short proof or a counterexample. (20)
i) Subtraction is a binary operation on N.
ii) If fv
1
; v
2
;:::; v
n
g is a basis for vector space V , fv
1 + v
2 + + v
n
; v
2
;:::; v
n
g is also
a basis for V .
iii) If W1
and W2
are subspaces of vector space V and W1 + W2 = V , then W1 \ W2 = f0g.
iv) The rank of a matrix equals its number of nonzero rows.
v) The row-reduced echelon form of an invertible matrix is the identity matrix.
vi) If the characteristic polynomial of a linear transformation is (x 1)(x 2), its
minimal polynomial is x 1 or x 2.
vii) If zero is an eigenvalue of a linear transformation T , then T is not invertible.
viii) If a linear operator is diagonalisable, its minimal polynomial is the same as the
characteristic polynomial.
ix) No skew-symmetric matrix is diagonalisable.
x) There is no matrix which is Hermitian as well as Unitary.