Last 10 notes on Perpetually Incomplete
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Extending the inner product from C^n to C^infinity and square-summable sequences, functional inner product spaces and L^2, the Fourier series basis of complex exponentials with orthogonality and projection formula, and convergence vs completeness.
The real dot product and its algebraic identities, the complex dot product with conjugated first argument, the axioms of a complex inner product space with proofs, and Dirac notation for inner products as a row-times-column product.
Normalising a vector to unit length, worked checks that two complex qubit states are unit vectors, the normalisation condition on amplitudes, and the definition of distance between two vectors.
The norm of real vectors as Euclidean length, its extension to generalised vectors, worked examples in R2 and R3, the norm of complex vectors with proof, and the Dirac notation for a vector's norm.
Worked practice on Dirac notation — determining matrix size and transpose, expressing a column vector in the computational basis, and computing the conjugate transpose (bra) of a complex vector.
Dirac notation for quantum state vectors — matrices, column and row vectors, transposition, kets and the computational basis, and the Hermitian adjoint (bras) with its conjugate-linear properties.
Coordinate vectors relative to a basis, the standard basis of R^n, and the definition of dimension and subspaces for finite-dimensional vector spaces.
Linear independence and redundant vectors, the definition of a basis, uniqueness of representation, and showing $\{(1,0,0), (1,1,0), (1,1,1)\}$ is a basis of $\mathbb{R}^3$.
Definition of a real or complex vector space and its axioms, subspaces and the closure test, and the span of a set of vectors — with $\mathbb{R}^2 = \operatorname{span}\{i, j\}$ as a worked example.
Vector equality, complex conjugates, addition and its properties, geometric representation (parallelogram and triangle rules), subtraction, scalar multiplication, and linear combinations in $\mathbb{R}^n$ and $\mathbb{C}^n$.