MHB Linear Transformation (Fredholm Alternative Theorem)

Swati
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Let T:V->V be a linear operator on an n-dimensional vector space. Prove that exactly one of the following statements holds:

(i) the equation T(x)=b has a solution for all vectors b in V.

(ii) Nullity of T>0
 
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Swati said:
Let T:V->V be a linear operator on an n-dimensional vector space. Prove that exactly one of the following statements holds:

(i) the equation T(x)=b has a solution for all vectors b in V.

(ii) Nullity of T>0

Hi Swati, :)

Suppose that the first statement is true. That is \(T\) is surjective. Then,

\[\mbox{dim }(V)=\mbox{dim }(\mbox{Im }T)=n\]

Then by the Rank-Nullity Theorem,

\[\mbox{Nullity }T=\mbox{dim }(\mbox{Ker }T)=0\]

Conversely you can show that if the second statement is true the first statement cannot be true.

Kind Regards,
Sudharaka.
 
how to proof if second statement is true then first statement is false.
 
Swati said:
how to proof if second statement is true then first statement is false.
Use the rank-nullity theorem again, to show that T is surjective.
 
if the SECOND statement is true, T CANNOT be surjective:

by the rank-nullity theorem:

dim(V) = rank(T) + nullity(T).

if nullity(T) > 0, then rank(T) < dim(V), so that:

dim(im(T)) < dim(V).

thus there is some b in V not in im(T).

(i only posted this because Opalg's post answers the wrong question).
 
Deveno said:
if the SECOND statement is true, T CANNOT be surjective:

by the rank-nullity theorem:

dim(V) = rank(T) + nullity(T).

if nullity(T) > 0, then rank(T) < dim(V), so that:

dim(im(T)) < dim(V).

thus there is some b in V not in im(T).

(i only posted this because Opalg's post answers the wrong question).
Yes of course. I should have said: Use the rank-nullity theorem again, to show that T is NOT surjective.
 
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