Thread: Basis for a vector space of Linear Mappings Question.

Hi eveyrone,

In class, we recently learnt that the set of all linear mappings that map vectors from the same domain to codomain is also a vector space. If it is indeed a vector space, then we can find a basis for a certain set.

I was wondering how we would find a basis for these mappings, or if there links that show an example of this. It is really confusing as to how we could express such a basis.

fill in the blanks here:

choose a basis for V and then find the ____ for a linear transformation. then find a basis for the set of _x_ _____s.

to pull this back to the set of all linear transformations, use the linear mapping corresponding to each basis ______ ,

which will be a basis for the set of all linear transformations V-->V.

(note: this only works for finite-dimensional vector spaces)

another, more direct approach: choose a basis {v1,v2,....,vn,....} for V (which may be finite, or infinite).

show that the set {Tij} where Tij(vk) = δjkvi, (where δjk is the kroenecker delta, δjj = 1, δjk = 0 j ≠ k) is a basis for the set of all linear maps V-->V.