This is probably a super simple question.
Let's say I have an equation
$$
\mathbf{v}^T(\mathbf{a} + \mathbf{b}) = \mathbf{v}^T(\mathbf{c}).
$$
Does this imply that
$$
(\mathbf{a} + \mathbf{b}) = \mathbf{c}
$$?
Intuitively, I would say that it does. On the other hand, there is no left hand side operation I can come up with that eliminates $\mathbf{v}$. I tried with:
$$
\mathbf{v} = \begin{pmatrix}v_1 \\ v_2\end{pmatrix}
$$
$$
\mathbf{x} = \begin{pmatrix}1/v_1\\1/v_2\end{pmatrix}
$$
And then
$$\mathbf{x}\mathbf{v}^T(\mathbf{a} + \mathbf{b}) = \mathbf{x}\mathbf{v}^T(\mathbf{c})
$$
But this doesn't result in the identity matrix on the left side as I had hoped:
$$
\begin{pmatrix}1/v1\\1/v2\end{pmatrix}\begin{pmatrix}v1\ v2\end{pmatrix} = \begin{pmatrix}1 \ \ v_2/v_1 \\ v_1/v_2 \ 1 \end{pmatrix}
$$
Best Answer
Generally speaking: no!
Imagine $\mathbf{v}$ being the zero vector. Or $\mathbf{v}$ being a vector both orthogonal to $\mathbf{c}$ and $\mathbf{a}+\mathbf{b}$.
A whole different story is the following:
$\mathbf{v}\cdot\mathbf{c}=\mathbf{v}\cdot(\mathbf{b}+\mathbf{a})$ for all $\mathbf{v}$ $\Rightarrow$ $\mathbf{c}=\mathbf{b}+\mathbf{a}$