Subspaces
Definition
W⊆ℝⁿ is a subspace if: (1) 0∈W, (2) closed under addition, (3) closed under scalar multiplication.
Examples
Example 1. Is the set of vectors with x+y=0 a subspace?
Solution. Yes — it's the null space of [1,1].
In Depth
A subspace of a vector space \(V\) is a subset \(W\subseteq V\) that is itself a vector space: it contains \(\mathbf{0}\), is closed under addition, and closed under scalar multiplication. Equivalently, \(W\) is closed under linear combinations.
The four fundamental subspaces of an \(m\times n\) matrix \(A\): column space \(\text{Col}(A)\subseteq\mathbb{R}^m\), null space \(\text{Null}(A)\subseteq\mathbb{R}^n\), row space \(\text{Row}(A)\subseteq\mathbb{R}^n\), left null space \(\text{Null}(A^T)\subseteq\mathbb{R}^m\). The column space and null space of \(A^T\) are orthogonal complements in \(\mathbb{R}^m\).
The span of a set \(S\) is the smallest subspace containing \(S\) — all linear combinations of elements of \(S\). The intersection of subspaces is a subspace; the union generally is not. The sum \(U+W=\{u+w:u\in U,w\in W\}\) is the smallest subspace containing both.
Affine subspaces (cosets) are translates of subspaces: \(\mathbf{x}_0+W=\{\mathbf{x}_0+w:w\in W\}\). The solution set of \(A\mathbf{x}=\mathbf{b}\) (when consistent) is an affine subspace: a particular solution plus the null space.
In abstract algebra, subspaces generalize to submodules (over rings) and ideals (in rings). In functional analysis, closed subspaces of Hilbert spaces are the natural setting for projection theorems and spectral theory. Every closed subspace of a Hilbert space has an orthogonal complement.
Key Properties & Applications
Krylov subspaces \(\mathcal{K}_k(A,\mathbf{b})=\text{span}\{\mathbf{b},A\mathbf{b},A^2\mathbf{b},\ldots,A^{k-1}\mathbf{b}\}\) are the basis of iterative methods for linear systems and eigenvalue problems. The Arnoldi process builds an orthonormal basis for \(\mathcal{K}_k\) and is the foundation of GMRES and the Lanczos algorithm.
Invariant subspaces: a subspace \(W\) is invariant under \(A\) if \(AW\subseteq W\). Eigenspaces are invariant. The Schur decomposition \(A=QTQ^*\) reveals a nested sequence of invariant subspaces (the columns of \(Q\) span invariant subspaces of increasing dimension).
In quantum mechanics, the state space is a Hilbert space and physical observables correspond to self-adjoint operators. Measurement collapses the state to an eigenspace of the observable. Superposition is a linear combination of states in different eigenspaces.
Further Reading & Context
The study of subspace connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.