Proving that $T:\mathbb R^N \rightarrow \mathbb R^N$ is not surjective

Question

Let $T:\Bbb R^n \rightarrow \Bbb R^n$ be a linear transformation and let $u_1,u_2$ different vectors in $R^n$ such that for every $v \in \Bbb R^n$ $Tu_1 \cdot v = Tu_2 \cdot v$.

Prove that $T$ is not surjective.

I'm clueless about where to begin, some help please?

Answer 1

$(Tu_1-Tu_2)\cdot v=0$ for every $v\in\Bbb R^n$, that is, $Tu_1-Tu_2$ is orthogonal to all vectors in the space, including itself.

What is the consequence?