If the statement is true, prove it. Otherwise give a counter example. a)If V=C3 and W1={(z1,z2,z2)∈C3:z1,z2∈C},...

If the statement is true, prove it. Otherwise give a counter example.

a)If V=C3 and W1={(z1,z2,z2)∈C3:z1,z2∈C}, W2={(0,z,0)∈C3:z∈C}, then V=W1⊕W2.

b)If Vis a vector space and W1, W2 are subspaces of V, then W1∪W2 is also a subspace of V.

c)If T:V→V is a linear operator, then Ker(T) and Range(T) are invariant under T.

d)Let T:V→V be a linear operator. If Ker(T)∩Range(T) ={0}, then V=Ker(T)⊕Range(T).

e)If T1,T2:V→V are linear operators such that T1T2=T2T1, and λ2 is an eigenvalue of T2, then Ker(T2−λ2I) is invariant under T1.

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Expert Solution

ekkarill92 answered 1 year ago

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