Show that $T$ is a linear operator - Linear Transformations in Linear Algebra We are asked:
Consider the operator $T:\mathbb{R^2} \rightarrow \mathbb{R^2}$ where $T(x_1, x_2) = (x_1 + kx_2, -x_2)$ for every $(x_1, x_2) \in \mathbb{R^2}$. Here $k \in \mathbb{R}$ is fixed.
a) Show that $T$ is a linear operator
b) Show that $T$ is one-to-one
We know in part (a) that a linear operator is one that satisfies the conditions of linearity
$$T(x_1 + x_2) = T(x_1) + T(x_2)$$
$$T(kx_1) = kT(x_1)$$
How do I apply that to the problem given?
We know that in part (b) a function is one-to-one whenever
$$T(x) = T(y)$$
for some $x$ and $y$ we must have $x=y$ (this is the property of 'onto').
I am almost certain that these responses are not proper solutions to my questions.

Hint:
there is some confusion between componets and vectors (or points) in your notation. 
For part a) you have to prove:
$$
T(\vec x+\vec y)=T(\vec x)+T(\vec y)
$$
with: $\vec x=(x_1,x_2)$ and $\vec y=(y_1,y_2)$
and 
$$
T(k \vec x)=kT(\vec x)
$$
with with: $\vec x=(x_1,x_2)$
Use the definition of T for $T \vec x$ and $T \vec y$ and you find the proof.
In the same way you can prove b).
