A really neat thing I learned in physics was Bra-Ket (get it bracket?) notation. Ket's are vectors and written like |u>. Bra's are objects that compute dot pro

A really neat thing I learned in physics was Bra-Ket (get it bracket?) notation. Ket's are vectors and written like |u>. Bra's are objects that compute dot products with vectors. They are written like <u|. So <u| |v> = u · v. However if you have |u> <v| then nothing happens. This is a new object. It will first take dot product with v and then multiply with |u> vector. Letting e1, e2.. denote the coordinate basis vectors, we can write a matrix A with coefficients a_ij as the sum Σa_ij |ej> <ei|. This can be rearranged as Σ_i |Σ_j a_ij ej> <ei| Written like this, it is clear that the column is where the unit vector ends up.