Question: Let Vbe a finite dimensional vector space and T:VV,S:VVbe two linear transformations such thatim(T)im(S) ={0}. Show thatker(T)ker(S) ={0}if and only ifV= im(T)im(S). Hint: 1. Use

Let Vbe a finite dimensional vector space and T:VV,S:VVbe two linear transformations such thatim(T)im(S) ={0}. Show thatker(T)ker(S) ={0}if and only ifV= im(T)im(S).

Hint:

1. Use the dimensions and all equalities you know about it.

2. The dimension of a subspace ofVis always smaller or equal thandim(V).)

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