I understand how to get eigenvalues, but I have a problem understanding eigenvectors.

I'll demonstrate with an example:

I am given A =

Solving for lambda, I get lambda = 1.5 and 4.5

Now my matrix equations become:

and

dividing by 1.5 I get:

so I get

Now, the answer is because if you set x_{1}=1, then x_{2}=1

If x_{1}=x_{2}, then why isn't the answer [ℝ ℝ], where ℝ is the set of all real numbers?