I read that a Linear Operator in Quantum Mechanics can be represented as A = $\sum_{ij} A_{ij}|u_{i}\rangle \langle u_{j}|$.
If A = $$ \begin{pmatrix} 3 & 2\\ 4 & 1 \end{pmatrix} $$, I used the formula above and got that A = $$\begin{pmatrix} 5 & 2\\ 0 & -2 \end{pmatrix}$$. I'm not sure what I'm doing wrong, since the basis I used is {[1/$\sqrt{2}$, 1/$\sqrt{2}$], [1/$\sqrt{2}$, -1/$\sqrt{2}$]}. What am I doing wrong? I also showed my work in the image attached.