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Recap. Complex vector space : a set of 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.

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recap
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.
recap continued
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.