Let endomorphism of a vector space of .Show that has no trivial null space iff has no trivial null space

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- September 8th 2007, 07:15 PMkezmanDual space problem
Let endomorphism of a vector space of .Show that has no trivial null space iff has no trivial null space

- September 8th 2007, 07:18 PMThePerfectHacker
- September 8th 2007, 07:29 PMkezman
Transpose of f

- September 8th 2007, 08:29 PMThePerfectHacker
I am not sure how "transpose" is defined for non-Euclidean linear transformations. But if V is an Euclidean space then can be tought as where is a square matrix. Since this endomorphism has no trivial null space, i.e. has non-trivial solutions it means . But since it means has non-trivial solutions. And so has the property mentioned.

- September 9th 2007, 09:57 AMkezman
Let K-vector spaces and let linear transf.

The function is defined:

- September 12th 2007, 10:20 PMRebesques
Suppose is non-trivial: there exists some such that . We are to find a nonzero functional such that ; Let be the length of the projection on (well defined). Then for all , we have or identically. So and is non-trivial.

For the converse, interchange between and .