MIT OpenCourseWare Linear Algebra Lecture: Eigenvalues, Invariant Subspaces, and Dirac Notation

Overview

MIT OpenCourseWare presents a lecture on linear algebra that revisits eigenvalues and invariant subspaces, then introduces inner products and the Dirac notation used in quantum mechanics. The instructor explains that restricting T to a one dimensional subspace U yields an eigenvalue equation, and that the spectrum of T consists of eigenvalues, possibly with degeneracies. He shows how the eigenvalue condition makes T−λI non-invertible and connects this to the determinant and the characteristic polynomial. The talk then shifts to inner product spaces, first in real space with the dot product, then in complex space where conjugation ensures a real norm. Dirac notation (bra-ket) is introduced as a convenient way to label vectors and act with bras to produce complex numbers. The session ends with a primer on orthogonality, Gram-Schmidt, orthonormal bases, and the notion of Hilbert spaces.

Introduction and Key Concepts

The lecture begins with a recap of invariant subspaces. A subspace U is T-invariant if T maps vectors in U back into U. When U is one-dimensional, generated by a nonzero vector u, the action of T on U must simply rescale u by some scalar λ, yielding the eigenvalue equation T u = λ u. This leads to the definition of eigenvalues and eigenvectors, and to the spectrum of T as the collection of eigenvalues, potentially with degeneracy.

The Spectrum and Characteristic Polynomial

The lecturer explains that the eigenvalue condition can be recast as (T − λI)u = 0, highlighting non-injectivity and non-invertibility. In a basis, this corresponds to the determinant of (T − λI) being zero, producing a polynomial f(λ) known as the characteristic polynomial. The zeros, which may be repeated, are the eigenvalues with possible degeneracy. The discussion contrasts real and complex vector spaces, noting that real matrices may lack real eigenvalues, while complex representations always reveal eigenvalues when working over the complex field.

Inner Products and Norms

The talk then introduces inner products as a map from V × V to the underlying field, motivating the familiar dot product in R^n and its key properties: commutativity, nonnegativity, and linearity in the second argument. The norm is defined via the inner product, with ||a||^2 = ⟨a, a⟩. The presenter notes that inner products need not be unique for a given vector space; different inner products can satisfy the same axioms, but standard choices are natural.

Complex Spaces and Dirac Notation

Extending to complex vector spaces requires a conjugate in one of the arguments to preserve positivity and to define the norm properly. The lecturer introduces the Dirac notation with bras and kets, where a ket |v⟩ is a vector in V and a bra ⟨w| is an element of the dual space V*, acting on vectors to yield complex numbers. The bra-ket formalism provides a convenient, basis-invariant way to express inner products and linear functionals. The duality between V and its dual is highlighted, including the one-to-one correspondence between vectors and corresponding bras in finite dimensions, and a concrete row-vs-column intuition for matrices.

Orthogonality, Orthonormal Bases, and Projections

With an inner product in place, one can define orthogonality: ⟨u, v⟩ = 0 means u and v are orthogonal. An orthonormal set satisfies ⟨e_i, e_j⟩ = δ_ij, providing a convenient basis for decomposition: any vector A can be written as A = ∑ a_i e_i, and the components a_i are given by a_i = ⟨e_i, A⟩. The norm induced by the inner product aligns with geometric length in Euclidean space. The Gram-Schmidt process is introduced as a method to convert a linearly independent set into an orthonormal basis, enabling clean projections and simple representations of vectors in the space.

Hilbert Spaces and Completeness

The lecture closes by differentiating finite and infinite dimensional spaces. In finite dimensions, inner product spaces are Hilbert spaces. For infinite dimensions, completeness must be ensured, which can be technical but is essential for quantum mechanics and functional analysis. The instructor emphasizes that plane waves and other non-normalizable constructs can be treated within generalized Hilbert spaces, while noting the physical meaning behind such mathematical tools.

Outlook

The session prepares students for further topics on linear operators, spectral theory, and more advanced inner product space theory in subsequent lectures.