Determining whether a transformation is linear I was hoping someone would be able to explain the following to me as I do not understand it:
f is the reflection in the line x = 1. 
The reflection f maps the origin to point (2, 0) so it does not fix the origin. Therefore f is not a linear transformation. 
Would someone be able to explain how f maps the origin to the point (2, 0) and why it is not linear. 
Thanks so much. 

Here is a visualization:

How does the reflection transformation map the origin?
$f(P) = P'$ such that the line segment $PP'$ is halved by the mirror line.
The shortest distance to the mirror line $x = 1$ is the same for point $P$ and mirrored point $P'$. 
Why is this transformation not linear?
Linear transformations have the properties
$$
T(x + y) = T(x) + T(y) \quad (x, y \in V) \\
T(\alpha x) = \alpha T(x) \quad (x \in V, \alpha \in F)
$$
for some vector space $V$ and field $F$.
If we use the second with $x = 0 = (0,0) \in \mathbb{R}^2$ and $\alpha = 2 \in \mathbb{R}$ we get
$$
T(2 \cdot 0) = 2\cdot T(0) \iff \\
T(0) = 2 \cdot T(0) \iff  \\
0 = T(0)
$$
So a linear map has to map the origin $0$ to itself.
