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I have a vector equation

$$\overrightarrow{x} = \langle c_1 + 3c_2, 17c_1 + c_2\rangle$$ where $c_1, c_2 \in \mathbb{R}$

Can I say that

$\overrightarrow{x} = \mathbb{R}^2$

Or I MUST have the vector equation $x = \langle a, b\rangle$ where $a, b \in R$ but independent of each other?

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    Asking whether $\vec x=\mathbb R^2$ does not make much sense. You have vector on one side and set of vectors on the other side of your equality. Did you mean $\vec x\in\mathbb R^2$? Or do you want to aske what are possible values of $\vec x$ (for arbitrary $c_{1,2}$)? – 2017-01-11

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Note that $x=\begin{bmatrix} 1 & 3 \\ 17 & 1 \end{bmatrix}\begin{bmatrix} c_1 \\ c_2 \end{bmatrix}$ and the fact that columns of matrix are independent so its rangespace is whole of $R^2$