Vector space Study Guide

📖 Core Concepts Vector space – a non‑empty set \(V\) with vector addition \((u,v)\mapsto u+v\) and scalar multiplication \((a,v)\mapsto av\) satisfying the eight axioms (closure, associativity, commutativity, zero vector, additive inverses, compatibility, identity, distributivity). Field \(F\) – the set of scalars (e.g., \(\mathbb R\) for real spaces, \(\mathbb C\) for complex spaces). Subspace – a non‑empty subset \(W\subseteq V\) closed under addition and scalar multiplication; \((W,+)\) is itself a vector space. Span – all finite linear combinations of a set \(S\); the smallest subspace containing \(S\). Basis – a linearly independent set whose span is the whole space; coordinates of any vector are the unique scalars in its linear combination of basis vectors. Dimension – cardinality of a basis; finite if a natural number, otherwise infinite. Linear map (homomorphism) – \(T:V\to W\) with \(T(u+v)=T(u)+T(v)\) and \(T(av)=aT(v)\). Isomorphism – bijective linear map; two spaces are isomorphic ⇔ they have the same dimension. Matrix representation – once bases of \(V\) and \(W\) are fixed, \(T\) corresponds to a unique matrix \(A\) with \([T(v)]W = A\,[v]V\). Determinant & invertibility – \(\det(A)\neq0\) ⇔ the associated linear map is an isomorphism (invertible). Eigenpair – non‑zero \(v\) with \(T(v)=\lambda v\); the set of all such \(v\) (plus \(0\)) is the eigenspace for \(\lambda\). Normed space – a vector space with a norm \(\|\cdot\|\) satisfying positivity, scalability \(\|av\|=|a|\|v\|\), and the triangle inequality. Inner product space – equipped with \(\langle\cdot,\cdot\rangle\) giving a norm \(\|v\|=\sqrt{\langle v,v\rangle}\) and satisfying linearity, symmetry (or conjugate symmetry), and positivity. Hilbert space – a complete inner‑product space; a basis means the closure of its span equals the whole space. --- 📌 Must Remember Vector‑space axioms (list them quickly): closure, associativity, commutativity, zero, additive inverse, scalar‑multiplication compatibility, identity \(1\cdot v=v\), distributivity over both vector and scalar addition. Dimension theorem: two finite‑dimensional spaces are isomorphic iff they have the same dimension. Rank‑nullity: \(\displaystyle \dim V = \operatorname{rank}(T)+\operatorname{nullity}(T)\). (Implicit from linear maps; useful for subspaces.) Determinant test: \(\det A\neq0 \iff A\) invertible \(\iff\) linear map is an isomorphism. Eigenvalue criterion: \(\lambda\) eigenvalue ⇔ \(\det(T-\lambda I)=0\). Gram–Schmidt: converts any linearly independent set \(\{v1,\dots,vk\}\) into an orthogonal (or orthonormal) set \(\{u1,\dots,uk\}\) by successive orthogonalization. Quotient space \(V/W\): elements are cosets \(v+W\); \(\dim(V/W)=\dim V-\dim W\) (finite‑dimensional case). --- 🔄 Key Processes Checking a Subspace Verify non‑emptiness (usually \(0\in W\)). Show closed under addition: if \(u,v\in W\) then \(u+v\in W\). Show closed under scalar multiplication: \(a u\in W\) for any \(a\in F\). Finding a Basis Start with a spanning set. Apply row‑reduction (or Gaussian elimination) to obtain a set of pivot columns → linearly independent. Those pivot vectors form a basis. Gram–Schmidt Orthogonalization Set \(u1 = v1\). For \(k\ge2\): \(uk = vk - \sum{j=1}^{k-1}\frac{\langle vk,uj\rangle}{\langle uj,uj\rangle}uj\). Optional: normalize \(ek = uk/\|uk\|\) for an orthonormal basis. Matrix of a Linear Map Choose basis \(\mathcal BV=\{v1,\dots,vn\}\) of \(V\) and \(\mathcal BW=\{w1,\dots,wm\}\) of \(W\). Compute \(T(vj)\) for each \(j\) and express in \(\mathcal BW\); column \(j\) of matrix \(A\) is the coordinate vector of \(T(vj)\). Eigenvalue/Eigenspace Computation Solve \(\det(T-\lambda I)=0\) for \(\lambda\). For each \(\lambda\), solve \((T-\lambda I)v=0\) to get the eigenspace. Constructing a Quotient Space Identify subspace \(W\). Form cosets \(v+W\). Define addition/scalar multiplication on cosets as in the outline. --- 🔍 Key Comparisons Vector space vs. Module Vector space: scalars form a field (every non‑zero scalar invertible). Module: scalars form a ring (may lack inverses); bases need not exist. Real vs. Complex vector space Real: scalars \(\mathbb R\); inner products are symmetric. Complex: scalars \(\mathbb C\); inner product is conjugate‑symmetric \(\langle u,v\rangle=\overline{\langle v,u\rangle}\). Subspace vs. Affine subspace Subspace: contains the zero vector, closed under addition and scalar multiplication. Affine subspace: translation \(x+V\) of a subspace; does not contain the origin unless the translation vector is zero. Basis (finite) vs. Hilbert‑space basis Finite: every vector is a finite linear combination of basis vectors. Hilbert: closure of linear span (including limits of infinite series) equals the whole space. Isomorphism vs. Similarity (matrices) Isomorphism: existence of a bijective linear map between spaces (dimension equality). Similarity: two matrices represent the same linear map in different bases; \(A = P^{-1}BP\). --- ⚠️ Common Misunderstandings “Any spanning set is a basis.” Missing linear independence; need both properties. “Zero vector can be an eigenvector.” Eigenvectors are non‑zero by definition; the zero vector always satisfies \(T(0)=\lambda 0\) trivially but is excluded. “Determinant zero ⇒ matrix not invertible ⇒ linear map not injective.” Correct, but remember the converse: non‑zero determinant guarantees both injective and surjective (isomorphism) for square matrices. “All modules have bases.” Only free modules have bases; many modules are non‑free. “Normed space automatically inner‑product space.” False; a norm need not arise from an inner product (e.g., \(\ell^1\) norm). --- 🧠 Mental Models / Intuition Vector space = “playground” where you can add arrows and stretch/compress them with scalars—everything obeys the same simple rules (the eight axioms). Basis = coordinate axes: just like \((x,y,z)\) in \(\mathbb R^3\), a basis lets you encode any vector as a list of numbers (coordinates). Quotient space = “collapsing” a subspace to a single point; think of gluing all vectors of \(W\) together, leaving cosets as new points. Eigenvectors = “directions that don’t turn” under a transformation; the transformation only scales them by \(\lambda\). Gram–Schmidt = “orthogonalizing” a messy set of directions step by step, like adjusting a set of skewed axes to become perpendicular. --- 🚩 Exceptions & Edge Cases Infinite‑dimensional spaces may lack a finite basis; concepts like dimension become cardinal numbers. Norms not induced by inner products (e.g., \(\|x\|1\) on \(\mathbb R^n\)). Non‑free modules have no basis; the structure theorem for finitely generated modules over a PID provides a decomposition instead. Determinant of non‑square matrix is undefined; invertibility only makes sense for square matrices. Quotient of infinite‑dimensional spaces can be infinite‑dimensional even if the subspace is large. --- 📍 When to Use Which To prove two vector spaces are “the same” → check dimensions; construct an explicit isomorphism if needed. To solve a system of linear equations → use row‑reduction to find a basis for the solution subspace (homogeneous) or a particular solution + nullspace (inhomogeneous). When you need orthogonal coordinates → apply Gram–Schmidt, especially in inner‑product or Hilbert spaces. To decide if a linear map is invertible → compute \(\det\) (square matrix) or check rank = dimension of domain. For spectral analysis (eigenvalues/eigenvectors) → use characteristic polynomial \(\det(T-\lambda I)=0\). If a problem involves “directions up to translation” → model with an affine subspace rather than a subspace. When working over a ring rather than a field → treat the structure as a module; avoid assuming existence of bases. --- 👀 Patterns to Recognize Zero vector appearing in a set → often indicates the set cannot be linearly independent. Closed under addition & scalar multiplication → hallmark of a subspace; check quickly in multiple‑choice. Determinant zero in a square matrix → expect non‑invertibility, non‑trivial kernel, possible eigenvalue \(0\). Characteristic polynomial factoring → each linear factor \((\lambda-\lambdai)\) signals an eigenvalue \(\lambdai\). Repeated columns/rows → hint at linear dependence ⇒ dimension drop. Inner product symmetry/positivity violation → not a valid inner product (common distractor). --- 🗂️ Exam Traps “All norms come from an inner product.” – False; only norms satisfying the parallelogram law are inner‑product‑induced. “If a set spans \(V\) then it is a basis.” – Missing independence; many spanning sets are larger than needed. Choosing the wrong basis for a matrix representation – forgetting to express \(T(vj)\) in the target basis leads to swapped rows/columns. Confusing eigenvectors with eigenvalues – answer choices may list a scalar where a vector is required. Assuming a quotient space \(V/W\) has dimension \(\dim V - \dim W\) for infinite‑dimensional spaces – the formula holds only when \(\dim V\) is finite. Misreading “orthogonal basis” as “orthonormal basis.” – orthogonal need not be unit length; extra normalization step required. Treating a module over a non‑field as a vector space – e.g., \(\mathbb Z\)-module \(\mathbb Z^n\) lacks scalar inverses, so concepts like basis behave differently. ---