Tell wheather the following are true or false give the reasonwhy true or counterexample why false: a) If V is a vector space other the the zero vector space. theV contains a subspace W such that W is not equal to V. b) An n x n diagonal matrix can never have more than n nonzeroentries. Tell wheather the following are true or false give the reasonwhy true or counterexample why false: a) If V is a vector space other the the zero vector space. theV contains a subspace W such that W is not equal to V. b) An n x n diagonal matrix can never have more than n nonzeroentries.
Explanation / Answer
(a) true. because , assuming W to the zero vector space , it is a proper subspace of every non zero vector space V. (b) a diagonal matrix is nothing but a square matrix whoseentries are zero other than the principal diagonal. while the matrix has n * n = n2 entries in which ndiagonal entries are not necessarily zero but all other arenecessarily zero, the matrix has n2 - n zero entriesatleast.