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

I'll demonstrate with an example:

I am given A = $\displaystyle \begin{bmatrix}3 & 1.5 \\1.5 & 3 \end{bmatrix}$

Solving for lambda, I get lambda = 1.5 and 4.5

Now my matrix equations become:

$\displaystyle -1.5x_1+1.5x_2=0$ and $\displaystyle 1.5x_1-1.5x_2=0$

dividing by 1.5 I get: $\displaystyle x_1-x_2=0$

so I get $\displaystyle x_1=x_2$

Now, the answer is $\displaystyle \begin{bmatrix}1 & 1 \end{bmatrix}$ 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?