# Unique linear transformations

1. Aug 24, 2005

### complexhuman

Unique linear transformations!!!

Problems agiain

Say I have 2 vector spaces with some finite number of vectors(can assume linear independency)...how can I show that the linear transformation between the two is unique?

2. Aug 24, 2005

### hello3719

Hint: translate the problem using a system of equations.

3. Aug 25, 2005

### HallsofIvy

Staff Emeritus
Perhaps I am misunderstanding something here. Do you mean that you are given a finite, linearly independent set of vectors in each space and the linear transformation must map each vector in one set into a corresponding vector in the other?

They way you have stated it, you can't prove the linear transformation is unique- it isn't. If the two sets are not bases for their vector spaces, then the linear transformation is not unique. Even if they are bases, unless you are requiring that the linear transformation map a specific vector in one set into a specific vector in the other then different linear transformations can map one set into the other, just rearranging which vector maps into which.

Assuming that you are given a basis for one space, in a specific order, and the linear transformation must map that into a basis for the other space, also in a given order, then you can show that the linear transformation is unique. One way would be to see what matrix represents that linear transformation in those bases.