1.Suppose that V is finite dimensional and T belong to L(V,W) .Prove
that T is surjective (onto) if only if there exists S belong to L(W, V
) such that  ST is the identity map on W

2. Suppose that W is finite dimensional and T belong to L(V,W) .Prove
that T is injective (onto) if only if there exists S belong to L(W, V
) such that ST is the identity map on W

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Request for Question Clarification by mathtalk-ga on 20 Feb 2003 06:09 PST

Hi, joannehuang-ga:

There seems to be a cut-and-paste error in your formulation, at least
of the second part of this problem.  While it is correct to identify a
linear transformation T as "surjective" if it is "onto", in the second
part the "injective" property (of T) would instead perhaps be
described as "1-1", certainly not as "onto".  I suspect you will want
to reverse the direction of composition of maps in the second part as
well.

Also your convention for the order of composition seems to be that:

 ST(v) = T(S(v)).

This convention is widely adopted in Europe but is contrary to how the
notation is often used in America.  It is good to define the notation
in any case, for the sake of rigor and clarity.

As a general suggestion for homework problems of this type, consider
how the underlying linear transformations S,T should act on a basis
for W,V respectively.

regards, mathtalk