I really need help with that question, so I'd appreciate a solution with steps because this problem was given for practice and my exam is in a few days so I'd like to understand the steps to the solution. Thank you.
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**HINT**

To spell out the comment hint a bit more, if $U$ spans $\mathbb{R}^2$, and $\vec{b} \in \mathbb{R}^2$, then there must exist $x,y,z \in \mathbb{R}$ such that
$$
\vec{b}
 = x\begin{bmatrix} 1\\2\end{bmatrix}
 + y\begin{bmatrix} 1\\3\end{bmatrix}
 + z\begin{bmatrix} 1\\4\end{bmatrix}
 = \begin{bmatrix} 1 & 1 & 1 \\2 & 3 & 4\end{bmatrix}
 \begin{bmatrix} x\\y\\z\end{bmatrix}
$$
So you have to prove that any $b$ can be generated that way...
For instance, $(1,3)$ is not a multiple of $(1,2)$, so they are independent and span $\Bbb R^2$.

But any $3$ vectors in a two-dimensional space are dependent, and so can't form a basis.

I have taken as given that $\Bbb R^2$ is two-dimensional, say with basis $e\_1=(1,0)$ and $e\_2=(0,1)$.