Question #45758

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.

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.

Expert's answer

Learn more about our help with Assignments: Engineering

## Comments

## Leave a comment