Tell wheather the following are true or false give the reasonwhy true or counter

Question

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.

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