Bra-Ket Notation and a Quantum Linear Algebra Review

Corresponds to Lectures 0.1–0.3 (asynchronous prerequisite material). A notational bridge rather than new mathematics: a ket $|\psi\rangle$ is a vector in a complex inner product space, a bra $\langle\phi|$ is the corresponding linear functional (via the Riesz representation / conjugate-linear duality), and $\langle\phi|\psi\rangle$ is their inner product — Dirac notation for exactly the objects covered in operators-on-inner-product-spaces. The special-operator classification (Hermitian, unitary, normal) and the spectral decomposition are the same spectral theorem material from that note, specialized to the complex case that quantum mechanics always uses; functions of operators (defined via the spectral decomposition, e.g. $e^{iHt}$) and the singular value decomposition round out the toolkit needed before the physical postulates in postulates-of-quantum-mechanics-and-the-qubit can be stated precisely.

Outline (TODO — flesh out each)

• Bras, kets, and linear operators — Dirac notation
• Special linear operators (Hermitian, unitary, normal) and the spectral decomposition
• Functions of operators; the singular value decomposition