Recap

1. Recap • Complex vector space: a setof abstract vectors or kets e.g. |v • closed under addition, multiplication by scalar complex numbers, i.e. all vectors you can reach via addition/scalar multiplication are elements of the vector space. • Zero vector plays a special role. • Linearly independent sets of vectors (l.i.v.) • sets where no member can be written as a linear sum of the others. • Dimension of a vector space: maximum number of l.i.v. that can co-exist in it. • Basis: set of l.i.v. big enough to allow any vector to be written as a sum over the basis vectors. Basis sets have to have the maximum size, i.e. one basis vector per dimension. • Coordinates of an arbitrary vector in a given basis: factors that multiply basis vectors |i in the linear sum: |a =  ai|i.

2. Recap (continued) • Inner products of vector pairs: e.g. a|b • give us orthogonality, norms, allow us to make orthonormal bases. • With orthonormal bases, coordinates are: ai = i|a • from coordinates can evaluate inner products, norms. • Left side of inner product seen as Bras, e.g. a| • live in dual of original vector space, e.g. row matrices c.f. kets as column matrices. Inner products then become ordinary matrix multiplication.