Linearity in the vector triple product I am trying to understand the vector triple product: 
$$x\times(y\times z).$$
As the vector triple product of $x$, $y$ and $z$ lies in the plane $(y \times z)$ the vector $x\times(y\times z)$ can be written as a linear combination of the vectors $\pm y$ and $\pm z$.
In the passage from the link below, can anyone please explain this sentence 

"As the left-hand side is a linear-function of $x$ so is the right-hand side and this means that $\alpha$ and $\beta$ are linear scalar valued functions of $x$".

Please show how the left-hand side is a linear function of $x$. What does linearity in vectors mean? Can anyone please explain in a bit of detail. Is there an online reference I can refer to?
Also, what does linear scalar valued functions of $x$ mean? Please explain Theorem $4.2.5$ here. To my understanding, this function changes a vector into a scalar. What does it mean to say $f : \mathbb{R}^3\to \mathbb{R}$ is linear. Please give examples.
 A: Let $V$ and $W$ be real vector spaces. A function $f : V \to W$ is linear if 


*

*$f(v_1 + v_2) = f(v_1) + f(v_2)$ for every $v_1, v_2 \in V$, and 

*$f(cv) = cf(v)$ for every $c \in \mathbb{R}$ and $v \in V$.


Saying that the expression $x\times(y\times z)$ is linear in $x$ means that for fixed $y, z \in \mathbb{R}^3$, the function $f : \mathbb{R}^3 \to \mathbb{R}$ given by $f(x) = x\times(y\times z)$ is linear. That is,


*

*$(x_1 + x_2)\times(y\times z) = x_1\times(y\times z) + x_2\times (y\times z)$ for all $x_1, x_2 \in \mathbb{R}^3$, and 

*$(cx)\times (y\times z) = c(x\times(y\times z))$ for all $c \in \mathbb{R}$ and $x \in \mathbb{R}^3$.


Both of these properties follow from whatever definition of cross product you are using.
A linear scalar valued function is a linear function $f : V \to \mathbb{R}$; i.e. a linear function which takes values in the scalars. 
In the case of Theorem $4.2.5$, $V = \mathbb{R}^3$ and the claim is that for any linear scalar valued function $f : \mathbb{R}^3 \to \mathbb{R}$, there is some $a \in \mathbb{R}$ such that $f(x) = x\cdot a$; i.e. every linear function $f : \mathbb{R}^3 \to \mathbb{R}$ is just given by the dot product with a fixed vector. 
Example: You can check that that the function $f : \mathbb{R}^3\to\mathbb{R}$ given by $f(x_1, x_2, x_3) = 6x_3-x_1$ is linear. So by Theorem $4.2.5$, there is some vector $a \in \mathbb{R}^3$ such that $6x_3-x_1 = x\cdot a$; in this case, $a = (-1, 0, 6)$.
