TITLE: surjective linear map from R to R²
QUESTION [5 upvotes]: How to quickly and clearly argue/show that there is or is not a linear surjective map $\phi$
$\phi: \mathbb{R} \to \mathbb{R}²$

REPLY [6 votes]: For a linear transformation in this case we have:
 1=dim R =dim ImT + dim KerT.
Then dim ImT is 0 or 1.