Let be a linear operator one the finite-dimensional space . Suppose there is a linear operator on such that = . Prove that is invertible and Give an example which shows that this is false when is not finite-dimensional.
THANKS
Let be a linear operator one the finite-dimensional space . Suppose there is a linear operator on such that = . Prove that is invertible and Give an example which shows that this is false when is not finite-dimensional.
THANKS
TU(v) = T(U(v)) = v
Hence range of T is entire V i.e. T is onto V.
As T is onto AND V is finite dimensional => T is invertible.
TU = I
(T^-1)TU = T^-1
U=T^-1
As for the example in infinite case - can't think. But I guess there was something like
T(q(x)) = q'(x)
U(q(x) = Integral from 1 to x q(x)
Please check it though
Here V is F[x] - all polynomials in x
Counterexample in the infinite dimensional case: let V be the real linear space of all infinite sequences and let T be the linear map defined by:
T({a_1, a_2,....,a_n,...}):= {a_2, a_3,....}
If U is defined as U({a_1, a_2,...}):= {0, a_1, a_2,...}, then we get
TU = I , but T isn't invertible since it is very not 1-1 (can you see it?)
Now just check both T, U are certainly linear maps.
Remark: if you prefer to make the example slightly fancier you may even choose V to be the l. s. of all convergent sequences or even all the convergent to zero sequences.
Tonio