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1 Vector spaces and dimensionality
LINEAR ALGEBRA: Vector spaces AND OPERATORS B. Zwiebach October 21, 2013 Contents 1 Vector spaces and dimensionality 1 2 Linear operators and matrices 5 3 Eigenvalues and eigenvectors 11 4 Inner products 14 5 Orthonormal basis and orthogonal projectors 18 6 Linear functionals and adjoint operators 20 7 Hermitian and Unitary operators 24 1 Vector spaces and dimensionality In quantum mechanics the state of a physical system is a Vector in a complex Vector space. Observables are linear operators, in fact, Hermitian operators acting on this complex Vector space. The purpose of this chapter is to learn the basics of Vector spaces , the structures that can be built on those spaces , and the operators that act on them. Complex Vector spaces are somewhat different from the more familiar real Vector spaces . I would say they have more powerful properties.
such case, no list of vectors from V can span V . Let us show that the vector space of all polynomials p(z) considered in Example 4 is an infinite dimensional vector space. Indeed, consider any list of polynomials. In this list there is a polynomial of …
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