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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?

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We say something is a vector if its transformation follows certain rules. This implies that we need to define transformation before we define the vector.

A Transformation transforms any three numbers from K to K' by such a rule that $T([x,y,z])=[x',y',z']$. We don't need the concept of vectors there.

To say $a$ is a vector means by knowing its representation in K, which is $[a_x,a_y,a_z]$, we can calculate its representation in K' by $T([a_x,a_y,a_z]) = [a_x',a_y',a_z']$.

To prove $a+b$ is a vector we need its two representations in K and K'. And if they follow the transformation $T$, it is a vector.

To get the representation of $a+b$ in K and K' we must define addition. You can define it like this:

  1. If you add two vectors you get a "thing" which has representations in different coordinate systems

  2. The representation of $a+b$ is such that in any coordinate system $(a+b)_x = a_x+b_x$, same with y and z.

Therefore you know $a+b$ has representations $[a_x+b_x,a_y+b_y,a_z+b_z]$ and $[a_x'+b_x',a_y'+b_y',a_z'+b_z']$ in K and K'.

Now if $T([a_x+b_x,a_y+b_y,a_z+b_z])=[a_x'+b_x',a_y'+b_y',a_z'+b_z']$ we can say $a+b$ is a vector.

Actually $T([a_x+b_x,a_y+b_y,a_z+b_z])=[(a_x+b_x)',(a_y+b_y)',(a_z+b_z)']$ and there's no rule that says $(a_x+b_x)' = a_x'+b_x'$.

That's where we add a condition that this transformation is linear, which gives exactly this $(a_x+b_x)' = a_x'+b_x'$ property.

This is kind of cheating but that's just mathematics, turning results you need into definitions.

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I think Feynman is making very heavy weather of this. Vectors can be defined by the rules of addition and scalar multiplication (in linear algebra we might also want to prove that they satisfy the axioms of vector space, which make no specific mention of coordinates). It is then straightforward to show that coordinate transformations are obeyed.

Physicists usually think of this a different way. They seek to abstract vectors as tabulated data from a physical situation. Then they use coordinate transformation as a test on the data to establish whether a particular tabulation represents a vector. An equivalent way of saying a particular tabulation is not a vector would be to say that we cannot combine members of that tabulation using addition and scalar multiplication to get another valid tabulation.

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A vector, in its geometric interpretation, is not merely a tuple of numbers. As Feynman says, a tuple is a vector if it satisfies a precise transformation law, when the coordinate system is changed. So let us assume that $\mathbf a$ and $\mathbf b$ are vectors. Under a coordinate transformation we can find a matrix $\Lambda$ such that the new components of the vectors in the new coordinate system are related to the old via the linear relation $$\mathbf a' = \Lambda \mathbf a,\qquad\mathbf b' = \Lambda\mathbf b.$$

If we now consider the sum $\mathbf a + \mathbf b$ merely as the tuple of its components, the question is whether this is a vector in the sense above. That is, is it true that, under the same transformation that gave us $\Lambda$, we have $(\mathbf a + \mathbf b)' = \Lambda(\mathbf a + \mathbf b)$? Given that $\Lambda$ is linear, this equality is satisfied precisely when $(\mathbf a + \mathbf b)' = \mathbf a' + \mathbf b'$. Therefore, provided we take for granted that

$$(\mathbf a + \mathbf b)' = \mathbf a' + \mathbf b'$$

the tuple of the components of $\mathbf a + \mathbf b$ transforms like a vector in the geometric sense.

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