I am reading the Feynman lectures on physics volume 1 chapter 11, Vectors. He says:
Addition of two vectors: suppose that $a$ is a vector which in some particular coordinate system has the three components $(a_x,a_y,a_z)$, and that $b$ is another vector which has the three components $(b_x,b_y,b_z)$. Now let us invent three new numbers $(a_x+b_x,a_y+b_y,a_z+b_z)$. Do these form a vector?
“Well,” we might say, “they are three numbers, and every three numbers form a vector.” No, not every three numbers form a vector! In order for it to be a vector, not only must there be three numbers, but these must be associated with a coordinate system in such a way that if we turn the coordinate system, the three numbers “revolve” on each other, get “mixed up” in each other, by the precise laws we have already described.
So the question is, if we now rotate the coordinate system so that $(a_x,a_y,a_z)$ becomes $(a_x′,a_y′,a_z′)$ and $(b_x,b_y,b_z)$ becomes $(b_x′,b_y′,b_z′)$, what does $(a_x+b_x,a_y+b_y,a_z+b_z')$ become? Do it become $(a_x′+b_x′,a_y′+b_y′,a_z′+b_z′)$ or not? The answer is, of course, yes, because the prototype transformations of Eq. (11.5) constitute what we call a linear transformation. If we apply those transformations to $a_x$ and $b_x$ to get $a_x′+b_x′$, we find that the transformed $a_x+b_x$ is indeed the same as $a_x′+b_x′$.
I am not able to understand this clearly. If $(a_x+b_x,a_y+b_y,a_z+b_z)$ represents a vector, then these three numbers will be transformed according to the rules described i.e. $x$ component on rotation through an angle $\theta$ becomes: $x\cos(\theta) + y\sin(\theta)$. But here we are finding out if $a_x + b_x$ is transformed the same way a vector component $x$ is transformed if it were subjected to rotation, but we cannot apply that transformation rule if we don't know that $a+b$ is a vector right? We want to prove that a component of $a+b$ transforms the same way as a component of $x$ (a vector), but he applies that transformation rule to $a_x+b_x$, and says the transformed $a_x$ plus $b_x$ is the same as $a_x$ individually transformed into $a_x’$ plus $b_x$ individually transformed into $b_x’$, ie $a_x’ +b_x’$. How is he able to apply the transformation rule if he wants to prove that its a vector, and doesn't know if its a vector, and what is a linear transformation?