Vector spaces and linear transformations

In short

A vector space is a set of objects that can be added together and scaled by numbers from a field, obeying eight axioms; a linear transformation is a map between two such spaces that preserves both operations. Together they are the backbone of linear algebra: every matrix is a linear map in disguise, and every system of linear equations is a question about a kernel or an image. B.Sc Algebra and Linear Algebra papers examine this heavily (definitions, proofs of independence, basis extension, rank-nullity), and it carries large weight in GATE Mathematics (MA), CSIR-UGC NET, IIT JAM, ISI/CMI entrances and TIFR screening. In placements it underpins data-science rounds at Mu Sigma, Fractal, Tiger Analytics, ZS Associates, quant desks at WorldQuant and Graviton, and ML/graphics roles at TCS Research, Qualcomm and Nvidia India.

Explained

Vector spaces: definition and examples

• A vector space V over a field F (usually ℝ, ℂ, or ℚ; also finite fields 𝔽_p) is an abelian group under addition together with scalar multiplication F × V → V.
• Eight axioms: (1) u+v = v+u; (2) (u+v)+w = u+(v+w); (3) a zero vector 0 with v+0 = v; (4) each v has an additive inverse −v; (5) a(u+v) = au+av; (6) (a+b)v = av+bv; (7) (ab)v = a(bv); (8) 1·v = v.
• Immediate consequences (prove them, they are asked): 0·v = 0, a·0 = 0, (−1)v = −v, and av = 0 ⟹ a = 0 or v = 0.
• Standard examples: ℝⁿ and ℂⁿ; M_{m×n}(F), all m×n matrices; P_n(F), polynomials of degree ≤ n; F[x], all polynomials; C[a,b], continuous real functions; the solution set of a homogeneous system Ax = 0; sequences; F^S, all functions from a set S to F.
• Non-examples: vectors in ℝ² with the first coordinate equal to 1 (no zero vector); the set of invertible matrices (not closed under addition); ℝ² with scalar multiplication a(x,y) = (ax, y) (axiom 6 fails: (a+b)v = ((a+b)x, y) but av+bv = (ax+bx, 2y)).
• The field matters: ℂ is a 1-dimensional vector space over ℂ but 2-dimensional over ℝ (basis {1, i}). ℝ over ℚ is infinite-dimensional.

Subspaces, span and sums

• Subspace test: a non-empty W ⊆ V is a subspace iff it is closed under addition and scalar multiplication; equivalently iff au + bv ∈ W for all a,b ∈ F and u,v ∈ W. Always check 0 ∈ W first — it is the fastest disproof.
• Intersection of any family of subspaces is a subspace; the union of two subspaces is a subspace only if one contains the other.
• Span: span(S) = set of all finite linear combinations a₁v₁ + … + a_k v_k; it is the smallest subspace containing S. By convention span(∅) = {0}.
• Sum: W₁ + W₂ = {w₁ + w₂}. It is direct (written W₁ ⊕ W₂) iff W₁ ∩ W₂ = {0}, equivalently every vector has a unique decomposition.
• Dimension formula for subspaces: dim(W₁ + W₂) = dim W₁ + dim W₂ − dim(W₁ ∩ W₂).
• Quotient space V/W: cosets v + W with (u+W)+(v+W) = (u+v)+W; dim(V/W) = dim V − dim W.

Linear independence, basis, dimension

• Vectors v₁,…,v_n are linearly independent if a₁v₁+…+a_n v_n = 0 forces every a_i = 0; otherwise dependent, and then some v_i is a combination of the others.
• Any set containing 0 is dependent; a single non-zero vector is independent; two vectors are dependent iff one is a scalar multiple of the other.
• A basis is an independent spanning set. Every vector then has a *unique* coordinate tuple with respect to it.
• Existence: every finitely generated vector space has a basis (extract one from a spanning set); in general, by Zorn's lemma, every vector space has a basis (Hamel basis).
• Replacement (Steinitz exchange) lemma: if S spans V with |S| = m and L ⊆ V is independent with |L| = n, then n ≤ m. Consequence: all bases have the same size, called dim V.
• Extension and reduction: any independent set extends to a basis; any spanning set contains a basis. In an n-dimensional space, any n independent vectors form a basis, and so does any n vectors that span.
• Standard dimensions: dim ℝⁿ = n, dim M_{m×n} = mn, dim P_n = n+1, dim(symmetric n×n) = n(n+1)/2, dim(skew-symmetric) = n(n−1)/2, dim(upper triangular) = n(n+1)/2, dim(trace-zero n×n) = n²−1.
• Practical test: put the vectors as rows of a matrix and row-reduce; the number of non-zero rows is the dimension of the span, and the vectors are independent iff that equals their count.

Linear transformations, kernel and image

• Definition: T : V → W is linear if T(u+v) = T(u)+T(v) and T(av) = aT(v); equivalently T(au+bv) = aT(u)+bT(v). Then T(0) = 0 always — a quick disqualifier for maps like T(x) = x + 1.
• Determined by a basis: if {v₁,…,v_n} is a basis of V, then arbitrary choices of T(v_i) ∈ W extend to exactly one linear map. This single fact answers a large fraction of exam questions.
• Kernel ker T = {v : T(v) = 0}, a subspace of V; image (range) Im T = T(V), a subspace of W. Nullity = dim ker T, rank = dim Im T.
• Injective ⟺ ker T = {0} (for linear maps only). Surjective ⟺ Im T = W. Bijective linear map = isomorphism; then T⁻¹ is also linear.
• Rank-nullity theorem: rank(T) + nullity(T) = dim V (the *domain*). Proof outline: take a basis {u₁,…,u_k} of ker T, extend to a basis {u₁,…,u_k,w₁,…,w_r} of V, and show {T(w₁),…,T(w_r)} is a basis of Im T.
• Corollary: if dim V = dim W (finite), then T injective ⟺ surjective ⟺ bijective. This fails in infinite dimensions: on F[x], the shift p(x) ↦ x·p(x) is injective but not surjective.
• Classification: two finite-dimensional spaces over the same field are isomorphic iff they have equal dimension; so every n-dimensional space over F is isomorphic to Fⁿ.
• The space L(V,W) of linear maps is itself a vector space with dim = (dim V)(dim W). The dual space V* = L(V,F) has dim V* = dim V, with the dual basis {f₁,…,f_n} defined by f_i(v_j) = δ_ij.
• Special maps: identity, zero, projection (T² = T, idempotent), involution (T² = I), nilpotent (T^k = 0), rotation and reflection in ℝ², differentiation on P_n (nullity 1, rank n), integration, trace, transpose.

Matrix of a linear map, and change of basis

• Fix ordered bases B = {v₁,…,v_n} of V and C = {w₁,…,w_m} of W. The matrix [T]_C^B is m×n, whose j-th column holds the coordinates of T(v_j) in C.
• Then [T(v)]_C = [T]_C^B [v]_B — matrix multiplication *is* composition of maps, and [S∘T] = [S][T].
• Change of basis: if P is the transition matrix from basis B′ to B (columns = coordinates of B′ vectors in B), then [v]_B = P[v]_{B′} and for an operator T : V → V, [T]_{B′} = P⁻¹ [T]_B P — the matrices are similar.
• Rank of the matrix = rank of the map, and it is independent of the bases chosen. Row rank = column rank.
• Four fundamental subspaces of A (m×n): column space ⊆ ℝ^m (= Im T), null space ⊆ ℝⁿ (= ker T), row space ⊆ ℝⁿ, left null space ⊆ ℝ^m; dim(row) = dim(col) = r, dim(null) = n − r, dim(left null) = m − r.
• Smith / rank normal form: any m×n matrix of rank r is equivalent to a block matrix [[I_r, 0],[0, 0]] — i.e. suitable bases make any linear map look like "copy r coordinates, kill the rest".
• Linear systems: Ax = b is consistent iff rank(A) = rank([A|b]); the solution set is then a coset x₀ + ker A, of dimension n − rank(A).

Inner product spaces and orthogonality

• An inner product ⟨·,·⟩ on a real space is bilinear, symmetric and positive definite; on a complex space it is conjugate-symmetric and linear in one argument. Norm ‖v‖ = √⟨v,v⟩.
• Cauchy-Schwarz: |⟨u,v⟩| ≤ ‖u‖‖v‖, with equality iff u, v are dependent. Triangle inequality follows. Parallelogram law: ‖u+v‖² + ‖u−v‖² = 2‖u‖² + 2‖v‖².
• Orthogonal set of non-zero vectors is always independent; an orthonormal basis makes coordinates trivial: v = Σ⟨v, e_i⟩e_i.
• Gram-Schmidt process turns any basis into an orthonormal one; it underlies the QR factorisation.
• Orthogonal complement W^⊥ is a subspace, V = W ⊕ W^⊥ (finite dimension), dim W + dim W^⊥ = dim V, and (W^⊥)^⊥ = W.
• Orthogonal projection of v onto W is the unique closest point of W to v — the basis of least-squares fitting.

Scope note and syllabus placement

• Under UGC CBCS / NEP-aligned B.Sc Mathematics, this material sits in the "Linear Algebra" core paper; AICTE's model curriculum places it in the first-year Mathematics courses for B.Tech. Exact paper codes and credits vary by university, so check your own scheme.
• Eigenvalues, diagonalisation and the Cayley-Hamilton theorem are examined as a *continuation* of this topic but are treated separately; determinants are usually a prerequisite chapter.

Key points

• Vector space - set with addition and scalar multiplication over a field F satisfying eight axioms; always verify 0 ∈ V.
• Subspace test - non-empty, closed under addition and scalar multiplication; equivalently au + bv stays inside.
• Span - set of all finite linear combinations; the smallest subspace containing the given set.
• Linear independence - the only vanishing linear combination is the trivial one.
• Basis - independent + spanning; coordinates with respect to it are unique.
• Dimension - common size of every basis; dim ℝⁿ = n, dim M_{m×n} = mn, dim P_n = n+1.
• Steinitz exchange - any independent set is no larger than any spanning set; this makes dimension well-defined.
• Dimension of a sum - dim(W₁+W₂) = dim W₁ + dim W₂ − dim(W₁∩W₂); direct sum iff the intersection is {0}.
• Linear map - preserves sums and scalars; completely determined by its values on one basis.
• Kernel and image - subspaces of the domain and codomain; injective iff kernel is zero.
• Rank-nullity - rank + nullity = dim(domain), *not* the codomain.
• Isomorphism - bijective linear map; two finite-dimensional spaces are isomorphic iff dimensions are equal.
• Matrix of a map - columns are images of the basis vectors written in the target basis; composition becomes multiplication.
• Change of basis - [T]_{B′} = P⁻¹[T]_B P; matrices of the same operator are similar and share rank, trace, determinant and eigenvalues.
• Row rank = column rank - a single number r controls solvability and solution dimension of Ax = b.
• Consistency - Ax = b solvable iff rank A = rank[A|b]; solution set is a translate of ker A.
• Orthonormal basis - coordinates are inner products; produced by Gram-Schmidt; gives V = W ⊕ W^⊥.
• L(V,W) and V* - dim L(V,W) = (dim V)(dim W); dim V* = dim V with dual basis f_i(v_j) = δ_ij.

Formulas and examples

(All quantities here are pure numbers — vector spaces are unitless; units enter only in applications such as least-squares fitting of measured data.)

Example 1 - Independence and basis of a span in ℝ⁴. Take v₁ = (1,2,0,1), v₂ = (2,4,1,3), v₃ = (0,0,1,1). Row-reduce:

• R2 → R2 − 2R1 gives (0,0,1,1); R3 is already (0,0,1,1).
• R3 → R3 − R2 gives (0,0,0,0).
• Non-zero rows: (1,2,0,1) and (0,0,1,1) → dim span = 2, so the three vectors are dependent, and indeed v₃ = v₂ − 2v₁ (check: (2,4,1,3) − (2,4,0,2) = (0,0,1,1) ✓).
• A basis of the span: {(1,2,0,1), (0,0,1,1)}. Extend to a basis of ℝ⁴ by adding e₂ = (0,1,0,0) and e₄ = (0,0,0,1) — the resulting 4×4 matrix row-reduces to the identity, so all four are independent.

Example 2 - Rank-nullity for an explicit map. T : ℝ⁴ → ℝ³, T(x,y,z,w) = (x + 2y − z, 2x + 4y + z + 3w, 3x + 6y + 3w). Matrix A = [[1,2,−1,0],[2,4,1,3],[3,6,0,3]].

• R2 → R2 − 2R1 = (0,0,3,3); R3 → R3 − 3R1 = (0,0,3,3); then R3 → R3 − R2 = (0,0,0,0).
• rank = 2, so nullity = 4 − 2 = 2, consistent with rank + nullity = 4.
• Kernel: from row 2, 3z + 3w = 0 ⟹ z = −w. From row 1, x + 2y − z = 0 ⟹ x = −2y + z = −2y − w. Free parameters y and w:

y = 1, w = 0 → (−2,1,0,0); y = 0, w = 1 → (−1,0,−1,1).

• Check the second: x+2y−z = −1 + 0 + 1 = 0 ✓; 2x+4y+z+3w = −2 + 0 − 1 + 3 = 0 ✓; 3x+6y+3w = −3 + 0 + 3 = 0 ✓.
• Image = span of two independent columns, e.g. column 1 (1,2,3) and column 3 (−1,1,0); T is neither injective nor surjective.

Example 3 - Matrix of a map and change of basis. T : P₂(ℝ) → P₂(ℝ), T(p) = p′ (differentiation), basis B = {1, x, x²}.

• T(1) = 0 → column (0,0,0); T(x) = 1 → column (1,0,0); T(x²) = 2x → column (0,2,0).
• [T]_B = [[0,1,0],[0,0,2],[0,0,0]] — strictly upper triangular, so T³ = 0 (nilpotent), rank 2, nullity 1 (the constants), matching 2 + 1 = 3 = dim P₂.
• Now take B′ = {1, 1+x, 1+x+x²}. Transition matrix P (columns = B′ vectors in B) = [[1,1,1],[0,1,1],[0,0,1]], with P⁻¹ = [[1,−1,0],[0,1,−1],[0,0,1]].
• [T]_{B′} = P⁻¹[T]_B P. First [T]_B P = [[0,1,1],[0,0,2],[0,0,0]]; then P⁻¹ times that = [[0,1,−1],[0,0,2],[0,0,0]]. Rank is still 2 — similarity preserves rank, trace (0) and determinant (0).

Example 4 - Dimension of a sum of subspaces. In ℝ⁴ let W₁ = span{(1,0,0,0), (0,1,0,0)} and W₂ = span{(0,1,0,0), (0,0,1,0), (0,0,0,1)}.

• dim W₁ = 2, dim W₂ = 3. Intersection: a vector in W₁ is (a,b,0,0); in W₂ it needs a = 0, so the intersection is span{(0,1,0,0)}, dimension 1.
• dim(W₁+W₂) = 2 + 3 − 1 = 4, so W₁ + W₂ = ℝ⁴ — but the sum is not direct, since the intersection is non-zero and 2 + 3 ≠ 4.

Example 5 - Gram-Schmidt in ℝ³. v₁ = (1,1,0), v₂ = (1,0,1), v₃ = (0,1,1).

• w₁ = (1,1,0), ‖w₁‖² = 2, e₁ = (1,1,0)/√2.
• ⟨v₂,w₁⟩ = 1 → w₂ = (1,0,1) − (1/2)(1,1,0) = (1/2, −1/2, 1); ‖w₂‖² = 1/4 + 1/4 + 1 = 3/2; e₂ = (1/2,−1/2,1)/√(3/2).
• ⟨v₃,w₁⟩ = 1, ⟨v₃,w₂⟩ = 0 − 1/2 + 1 = 1/2 → w₃ = (0,1,1) − (1/2)(1,1,0) − (1/2)/(3/2)·(1/2,−1/2,1) = (0,1,1) − (1/2,1/2,0) − (1/6,−1/6,1/3) = (−2/3, 2/3, 2/3).
• Check orthogonality: ⟨w₃,w₁⟩ = −2/3 + 2/3 = 0 ✓; ⟨w₃,w₂⟩ = −1/3 − 1/3 + 2/3 = 0 ✓. Then e₃ = (−1,1,1)/√3.

Where it is used

• Data science and analytics (Mu Sigma, Fractal, Tiger Analytics, ZS Associates, Bengaluru/Gurugram): feature matrices, rank and multicollinearity checks, PCA and least-squares regression, all stated as projections onto column spaces.
• Machine learning and computer vision (TCS Research, Wipro AI, Qualcomm India, Nvidia Bengaluru): embeddings are vectors in high-dimensional spaces; every neural network layer is a linear map followed by a non-linearity.
• Computer graphics and gaming (Ubisoft Pune, Rockstar India, Dhruva/Technicolor): rotations, scalings, shears and projections implemented as 4×4 matrices; change of basis is the camera transform.
• Quantitative finance (WorldQuant, Graviton, Tower Research, Mumbai/Bengaluru): portfolio returns as linear combinations, covariance matrices, factor models and hedging as projections.
• Engineering and ISRO/DRDO work: finite element analysis assembles stiffness matrices; control systems use state-space models ẋ = Ax + Bu; signal processing uses Fourier and wavelet bases, which are orthonormal bases of function spaces.
• Coding theory and cryptography: linear codes are subspaces of 𝔽_qⁿ with the generator matrix as basis and the parity-check matrix defining a kernel — used in telecom standards and in CSIR-NET/GATE questions.

Interview questions

• Q: What makes a subset a subspace, and what is the fastest way to disprove it? A: It must be non-empty and closed under addition and scalar multiplication; the fastest disproof is showing the zero vector is absent.
• Q: Can the union of two subspaces be a subspace? A: Only if one is contained in the other; otherwise take u from one and v from the other and u+v lies in neither.
• Q: State the rank-nullity theorem and name the common error. A: rank(T) + nullity(T) = dim(domain); the common error is using the dimension of the codomain instead.
• Q: If T : ℝ⁵ → ℝ³ is linear, can it be injective? A: No — nullity = 5 − rank ≥ 5 − 3 = 2 > 0, so the kernel is non-trivial.
• Q: Why is a linear map completely determined by its action on a basis? A: Every vector is a unique linear combination of basis vectors, and linearity forces the image of that combination.
• Q: Why do row rank and column rank agree? A: Row operations preserve the row space and the dependence relations among columns, so reduction to row echelon form exhibits the same number r for both; equivalently rank(A) = rank(Aᵀ).
• Q: What does it mean for two matrices to be similar, and what is preserved? A: B = P⁻¹AP — they represent the same operator in different bases; rank, trace, determinant, characteristic polynomial and eigenvalues are preserved.
• Q: Give a linear map that is injective but not surjective. A: On the infinite-dimensional space F[x], p(x) ↦ x·p(x); impossible in finite dimensions when domain and codomain have equal dimension.
• Q: Is T(x, y) = (x + 1, y) linear? A: No — T(0,0) = (1,0) ≠ 0, and linear maps must send zero to zero.
• Q: What is the dimension of the space of all 3×3 symmetric matrices? A: 6, namely n(n+1)/2 with n = 3 — three diagonal and three off-diagonal independent entries.
• Q: Why is an orthogonal set of non-zero vectors automatically independent? A: Take a vanishing combination and inner-product it with each vector; all cross terms vanish, forcing every coefficient to be zero.
• Q: When is Ax = b solvable, and what does the solution set look like geometrically? A: When rank A = rank[A|b]; it is then a translate (coset) of the null space, an affine subspace of dimension n − rank A.

Common mistakes

• Using the dimension of the codomain in rank-nullity instead of the domain.
• Forgetting to check that 0 belongs to a claimed subspace, or testing only closure under addition.
• Assuming the union of subspaces is a subspace, or that any sum of subspaces is direct.
• Confusing "spanning" with "independent" — a spanning set can be too big, an independent set too small; a basis is both.
• Writing the matrix of a map with the images of basis vectors as *rows* instead of columns, or mixing up P and P⁻¹ in the change-of-basis formula.
• Claiming injective ⟺ surjective without the hypotheses of equal finite dimension.
• Treating a non-homogeneous solution set as a subspace — it is a coset, and contains 0 only when b = 0.
• Carrying finite-dimensional intuition into function spaces, where a basis may not be computable and operators need not be bounded.

Related topics

Matrices, eigenvalues and diagonalisation · Group theory · Rings and fields · Determinants and systems of linear equations · Inner product spaces and quadratic forms · Numerical linear algebra and least squares