# Answer to Question #42312 in Linear Algebra for Anurodh

Question #42312

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

i) Subtraction is a binary operation on N.

ii) If{ v1, v2,..., vn} is a basis for vector space V,{ v1+ v2+···+ vn, v2, ..., vn} is also a basis for V.

iii) If W1 and W2 are subspaces of vector space V and W1+W2 =V, then W1∩W2 ={0}.

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.

i) Subtraction is a binary operation on N.

ii) If{ v1, v2,..., vn} is a basis for vector space V,{ v1+ v2+···+ vn, v2, ..., vn} is also a basis for V.

iii) If W1 and W2 are subspaces of vector space V and W1+W2 =V, then W1∩W2 ={0}.

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.

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