I have a set of vectors $\mathbf{V} = [\mathbf{v}_{1} \cdots \mathbf{v}_{M}] \in \mathbb{R}^{N \times M}$. The sum of these vectors will form another vector $\mathbf{w} \in \mathbb{R}^{N}$, as is already known.
Now, say that for each $\mathbf{v}_{m}$, I apply a scaling and rotation factor $\alpha_{m}$ and $\mathbf{R}_{m} \in \mathbb{R}^{N \times N}$, the sum will give me a vector $\tilde{\mathbf{w}}$ which is possibly different from $\mathbf{w}$.
I don't know how to prove it (hence the question), but from some simulations, it is clear that $\tilde{\mathbf{w}}$ is a scaled and rotated version of $\mathbf{w}$ (see Fig. 1).
We have $\mathbf{w} = \sum_{m=1}^{M} \mathbf{v}_{m}$ and $\tilde{\mathbf{w}} = \sum_{m=1}^{M} \alpha_{m}\mathbf{R}_{m}\mathbf{v}_{m}$.
If we suppose that $\alpha_{m} = \gamma_{m}\alpha_{1},~m \neq 1$ and $\mathbf{R}_{m} = \mathbf{S}_{m}\mathbf{R}_{1},~m \neq 1$, we can reduce the expression to $\tilde{\mathbf{w}} = \alpha_{1}\mathbf{R}_{1}\Big(\mathbf{v}_{1} + \sum_{m=2}^{M} \gamma_{m}\mathbf{S}_{m}\mathbf{v}_{m}\Big)$, which is still not entirely a simple scaling and rotation of the original basis $\mathbf{V}$ (this is where I am at currently).
I wonder if there is a way to prove this. The objective for me is not to obtain a direct relation between the different parameters, but rather to prove simply that $\tilde{\mathbf{w}}$ may be characterised on the overall, as a 'total' scaling and rotation of $\mathbf{w}$.
Fig.1 Example of applying scale and rotation to vectors. Black: original vectors in 3-D Euclidean space. Red: scaled and rotated versions of the original. Broken lines: individual vectors. Full line: vector formed from the sum of each individual vectors
P.S. The trivial case is when $\forall~m,~\alpha_{m} = \gamma$ and $\forall~m,~\mathbf{R}_{m} = \mathbf{S}$, in which case we can write $\tilde{\mathbf{w}} = \gamma\mathbf{S}\mathbf{w}$.
Two arbitrary vectors can always be related to each other by a scaling and a rotation. In particular, the scaling factor is the ratio of the norms; in 3D, the rotation axis is parallel to the cross product, and in $N$D it is not uniquely defined.
Your question is somewhat vacuous.