States can be represented as vectors

|Ψ> = ½|+> + (i√3)/2|-> =  ½ ( 1 ) + (i√3)/2 ( 0 ) = (   1/2    )
      	     		     ( 0 )   	     ( 1 )   ( (i√3)/2  )

Operators are matrices

Showed how Â|Ψ>=|φ> translates to a matrix in the z basis (pic x2)

Example with S𝓏 operator (pic)
Example with S𝓍 operator (pic)

Often we know operator but not basis. How to find basis? (pic)
      i.e. find the eigen-basis S|α>=λ|α>
      Only non-trivial if determinant is zero. Solve for basis. (pic)
