Transcription of 1 Vector spaces and dimensionality
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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.
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 maximum degree (recall the list is finite). Thus polynomials of higher degree are not in the span of the list.
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