2 questions : Kernel/Image of a linear transformation and orthonormal basis

**Hello MHF**,

It is the continuation of the exercise I posted in my precedent thread.

So I recall, I have a vectorial subspace of , . is defined to be .

I've found a basis of and a basis of . They now ask me to find an orthonormal basis of using the basis I found in the past items.

For I have that a basis is . While for I have that a basis is .

Oh wait... doesn't that mean that a basis of is the union of the 2 basis I found? But it wouldn't necessarily be orthonormal especially because I have a factor 2 in the basis of . So I don't see how I can answer the question.

Now the next question seems even much harder. I must find a linear transformation such that and .

My attempt : and I'm at a loss.