Part of forming a basis is that the vectors u,v,w have to be linearly independent. That means that each of the vectors can't be a linear combination of the others. So, let's look at the new vectors (let's call them x,y,z)
x = u+v
y = -u+w
z = u+v+w
If you take any of x,y and z, and linearly combine them, can you end up with another one of x, y or z (hint: you can't)? If not, x, y and z form a basis.
As for the second part of the question, are we talking affine transformation? Those are most easily expressed in a matrix, which reduces the transform to a vector-matrix multiply (which itself is just a series of dot products).
I'm not sure what formulae are being looked for here, because it depends on the transform asked for. If it's just a rotation, it's relatively simple - you can just build a matrix like so:
\[\left[\begin{matrix}\cos \beta \cos \gamma & -\cos \alpha \sin \gamma + \sin \alpha \sin \beta \cos \gamma & \sin \alpha \sin \gamma + \cos \alpha \sin \beta \cos \gamma\\ \cos \beta \sin \gamma & \cos \alpha \cos \gamma + \sin \alpha \sin \beta \sin \gamma & -\sin \alpha \cos \gamma + \cos \alpha \sin \beta \sin \gamma \\ -\sin \beta & \sin \alpha \cos \beta & \cos \alpha \cos \beta\end{matrix}\right]\]
where
\[\alpha, \beta, \gamma\]
are Euler angles (rotations about the X, Y, and Z axes of your coordinate system), respectively.
A multiplication of a 3-vector with such a matrix would yield a 3-vector rotated by the three angles. It follows, that for example to get the transformed x-component of the vector v [xyz], you would calculate
\[x' = x*cos \beta cos \gamma + y*cos \beta sin \gamma + z*-sin \beta \]
And do similarly for the y and z components with the second and third column of the matrix.
Someone please double check my math here, it's been a while since I've actually built a rotation matrix by hand :P