1 Vector spaces and dimensionality - MIT OpenCourseWare

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 .

3. There is a vector 0 ∈ V such that 0+ u = u for all u ∈ V (additive identity). 4. For each v ∈ V there is a u ∈ V such that v + u = 0 (additive inverse). 5. The element 1 ∈ F satisfies 1v = v for all v ∈ V (multiplicative identity). 6.

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