Recap

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Recap - PowerPoint PPT Presentation

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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Presentation Transcript
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)
• 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.