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 $\displaystyle \mathbb{R}^4$, $\displaystyle W$. $\displaystyle W$ is defined to be $\displaystyle \{ (x,y,z,w)\in \mathbb{R}^4 \text{such that }2x-z+w=0, x-y-z=0 \}$.

I've found a basis of $\displaystyle W$ and a basis of $\displaystyle W^{\perp}$. They now ask me to find an orthonormal basis of $\displaystyle \mathbb{R}^4$ using the basis I found in the past items.

For $\displaystyle W$ I have that a basis is $\displaystyle \{ (0,1,-1,-1), (1,0,1,-1) \} $. While for $\displaystyle W^{\perp}$ I have that a basis is $\displaystyle \{ (2,0,-1,1), (1,-1,0,-1) \}$.

Oh wait... doesn't that mean that a basis of $\displaystyle \mathbb{R}^4$ 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 $\displaystyle W^{\perp}$. 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 $\displaystyle T:\mathbb{R}^4 \to \mathbb{R}^4$ such that $\displaystyle \ker (T)=W$ and $\displaystyle \Im (T)=W^{\perp}$.

My attempt : $\displaystyle T(x,y,z,w)=0\Leftrightarrow T(x,y,x-y,-x-y)=0$ and I'm at a loss.