Question

Let V be a vector space and define T:V-> V by T() = a* where a is any non-zero scalar. Show that T is a linear transformation. Then show that T is one-to-one and onto and find T^(-1) (T inverse)

Answer 1

This is a test question, I have a good idea of how to solve it. Just wanted someone to collaborate with and check my answers.

Answer 2

It's pretty straight forward isn't it? There are a number of axioms for T to be a vector space linear transformation. Check them all, they all fall out very easily.

Now, provided a isn't a non-zero constant, then you can guess the form of the inverse.

Answer 3

That's what I figured. The transformation is clearly linear, so I wont worry about that. It's the kernel that I'm unsure of. Since a is non-zero I stated:
Kern(T) = T(<v>) = <0>
a*<v> = <0>
<v> = <0>
(Implies 1-1)

Rng(T) = T(<v>) = <w>
a*<v> = <w>
where our range is the space spanned by 1/a<w>
and T inverse is (1/a)

Answer 4

Yes, it clearly has the trivial kernel and the range is the entire space. T inverse is
T(v) = (1/a)v. 1/a exists provided a is not zero.