I want to compute the Householder-transformation that maps the first column of $ A = \begin{bmatrix} 0 & 5 \\ 2 & 3 \\ \end{bmatrix} $ to a multiple of $e_1$.
Using that $v_1 = \begin{bmatrix} 0 \\ 2 \\ \end{bmatrix}$, $u_1 = v_1 + ||v_1||e_1$ and $H_1 = I_2 - $$2u_1u_1^T \over u_1^Tu_1$, I get that $H_1 \begin{bmatrix} 0 \\ 2 \\ \end{bmatrix} = \begin{bmatrix} -16 \over 10 \\ 6\over10 \\ \end{bmatrix}$. I just don't see what I did wrong.
A Householder transformation is a reflection about a (hyper)plane.
If you have a vector ${\boldsymbol v}$, then applying the Householder transformation on it gives you a vector ${\boldsymbol v}'$ which is its reflection about a (hyper)plane defined by a unit-norm vector ${\boldsymbol n}$ (normal to the (hyper)plane) and we write: $$ {\boldsymbol v}'={\boldsymbol v}-2{\boldsymbol n}{\boldsymbol n}^T{\boldsymbol v} \tag{1} $$
Now, if you want your ${\boldsymbol v}=[0,2]^T$ to be reflected on the x-axis (i.e. a multiple of ${\boldsymbol e}_1$), the hyperplane (line in 2D) about which you want your reflection is the hyperplane defined by vector ${\boldsymbol n}=[-\sqrt{2},\sqrt{2}]^T$ (see figure below).
What you did wrong in your question was that you used a vector ($u_1$) that is parallel to the hyperplane instead of taking a vector which is normal to it.
You can plug in the numbers in $(1)$ and check that ${\boldsymbol v}'=[2,0]^T$.