Give an example of two linear maps $T_1$ and $T_2$ from $\mathbf R^5$ to $\mathbf R^2$ that have the same null space but are such that $T_1$ is not a scalar multiple of $T_2$.

Let $T_1(x_1,\dots,x_5) = x_1+x_2$ and $T_2(x_1,\dots,x_5) = x_1-x_2$. Clearly

But $T_1 \ne \lambda T_2$ for any $\lambda \in \mathbf R$ since $x_1+x_2$ cannot be made to equal $\lambda(x_1-x_2)$ at every $x_1,x_2$.