Press Ctrl+P (or ⌘+P) and choose Save as PDF. Free to keep and to pass on.

CSIR NET Mathematical Sciences · revision sheet

Analysis & Linear Algebra

29 subtopics · roughly 60 marks · 35 counterexamples

What the exam asks of each subtopic

The Real Line

Completeness, sup/inf, Archimedean property.: Every question here is really 'which property of ℝ fails in ℚ'. Completeness (sup exists) is what separates them; Archimedes and density follow from it.
Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy.: Bolzano–Weierstrass and the Cauchy criterion are tested as *decisions*: given a sequence, is it bounded / Cauchy / convergent / does it have a convergent subsequence? Know exactly which implications hold in ℝ and which need completeness.
limsup, liminf and subsequential limits.: Compute limsup/liminf of explicit sequences fast, and know the algebra: limsup(aₙ + bₙ) ≤ limsup aₙ + limsup bₙ, with equality if one converges. Convergence ⇔ limsup = liminf (finite).
Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements.: Know which test to reach for in 10 seconds, the limsup forms, and the three classic traps: conditional vs absolute convergence, rearrangements (Riemann), and 'terms → 0 does not imply convergence'.

Continuity and Differentiation

Continuity, uniform continuity, Lipschitz.: The chain Lipschitz ⇒ uniformly continuous ⇒ continuous, where each converse fails, and what compactness or boundedness of the domain changes. Know √x, 1/x, x², sin(1/x), sin(x²) by heart.
Differentiability, mean value theorems, Taylor, L'Hôpital.: MVT-family questions are about *which hypothesis is missing*: Rolle needs continuity at the endpoints, Darboux needs only differentiability, L'Hôpital needs the derivative-quotient limit to exist. Know x² sin(1/x) cold.

Integration

Riemann integration and criteria.: Lebesgue's criterion (bounded + discontinuities of measure zero) decides Riemann integrability instantly. Know Dirichlet, Thomae, and that composition can destroy integrability.
Improper integrals and convergence tests.: Split at every singular point and at infinity, compare with 1/xᵖ, and distinguish convergence from absolute convergence — ∫ sin x/x is the standard example of the gap.

Sequences and Series of Functions

Pointwise vs uniform convergence, M-test, Dini.: Uniform convergence is what lets you swap limit with integral/derivative/continuity. Compute sup|fₙ − f| explicitly; xⁿ, nx(1−x)ⁿ, nxe^{−nx²}, x/n are the recurring families.
Power series, radius of convergence, Abel's theorem.: Radius via limsup |aₙ|^{1/n} (Cauchy–Hadamard), never assume the ratio limit exists. Behaviour on the boundary circle is decided separately (Abel); differentiation/integration keep the radius.
Arzelà–Ascoli and equicontinuity.: Arzelà–Ascoli = uniformly bounded + equicontinuous ⇒ a uniformly convergent subsequence. It is the compactness criterion in C[a,b]; the failures are xⁿ (not equicontinuous) and constants n (not bounded).

Functions of Several Variables

Partial derivatives, differentiability, chain rule.: Partials existing ⇏ continuous ⇏ differentiable. The safe implication is: continuous partials ⇒ differentiable. Know xy/(x²+y²) and the equality-of-mixed-partials failure.
Inverse and implicit function theorems, extrema.: Both theorems need a non-vanishing Jacobian (∂F/∂y ≠ 0 for implicit). Where it vanishes, anything can happen — that is exactly what the questions probe.

Metric Spaces

Open/closed sets, limit points, closure, interior.: Sets can be both open and closed, or neither. Know which operations preserve openness (arbitrary unions, finite intersections) and the standard ℚ examples.
Compactness: open covers, sequential, Heine–Borel.: Closed + bounded ⇒ compact ONLY in ℝⁿ. Know the ℓ² unit ball and discrete ℝ as spoilers. Compact ⇒ complete ⇒ closed.
Completeness and Baire category.: Completeness: closed subsets of complete spaces, ℓᵖ, C[0,1] with sup norm are complete; ℚ, (0,1), C[0,1] with L¹ norm are not. Baire: ℝ is not a countable union of nowhere dense sets; a complete metric space without isolated points is uncountable.
Connectedness and path-connectedness.: Connected subsets of ℝ are intervals; continuous images stay connected; path-connected ⇒ connected with the topologist's sine curve as the standard converse failure — except open subsets of ℝⁿ, where the two agree.

Lebesgue Measure and Integration

Measurable sets and functions.: Measure zero, countable additivity and 'almost everywhere' are the workhorses. Know that measurable ⊋ Borel and that a non-measurable set requires the axiom of choice.
Lebesgue integral, MCT, DCT, Fatou.: Pick the right convergence theorem: MCT (increasing, non-negative), Fatou (inequality, always), DCT (needs a dominating integrable g). The moving-bump examples show what happens without domination.
L^p spaces essentials.: L^p inclusions go one way on finite measure spaces and the other way for ℓ^p — getting the direction right is most of the battle.

Vector Spaces and Linear Maps

Bases, dimension, rank–nullity.: Rank–nullity plus 'rank is unchanged by field extension' answers most of these. Watch the direction of the rank inequalities for products.
Linear transformations, matrix representation, change of basis.: Similar matrices are the same operator in different bases: they share rank, trace, determinant, characteristic and minimal polynomials — but sharing those is not enough to be similar.

Eigenvalues and Canonical Forms

Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton.: Read eigen-structure off the characteristic and minimal polynomials without computing: diagonalisable ⇔ minimal polynomial has distinct linear factors; similar matrices share both polynomials but the converse fails; Cayley–Hamilton lets you compute inverses and high powers.
Diagonalisability criteria.: Decide diagonalisability from a single equation or property: idempotent, involution, Aᵏ = I, nilpotent, symmetric, normal, distinct eigenvalues. Know over which field.
Jordan canonical form.: Recover the block structure from three numbers per eigenvalue: algebraic multiplicity (total size), geometric multiplicity (number of blocks), minimal-polynomial exponent (largest block).
Rational canonical form.: Rational canonical form works over any field — use it when the characteristic polynomial does not split. Invariant factors divide one another; the last one is the minimal polynomial.

Inner Product Spaces and Forms

Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem.: Spectral theorem: real symmetric ⇒ orthogonally diagonalisable with real eigenvalues; complex normal ⇒ unitarily diagonalisable. Know which matrix classes are normal.
Quadratic forms, positive definiteness, Sylvester's law.: Signature is the complete invariant over ℝ (Sylvester). Positive definiteness is tested by leading principal minors, or by all eigenvalues being positive — do not mix the two tests up.

Determinants and Matrix Tricks

Determinants, trace, block matrices, rank inequalities.: Block formulas and the trace/determinant identities turn hard computations into one-liners. det(I + AB) = det(I + BA) is the single most useful trick here.

Tempting, and false

Each claim below feels true and is not. The object beside it is the one that settles it.

Cesàro means converge ⇒ the sequence converges
↳ aₙ = (−1)ⁿ — Partial averages → 0, sequence diverges.
limsup(aₙ + bₙ) = limsup aₙ + limsup bₙ
↳ aₙ = (−1)ⁿ, bₙ = (−1)ⁿ⁺¹ — aₙ + bₙ ≡ 0, so the left side is 0 while the right side is 1 + 1 = 2.
aₙ → 0 ⇒ Σaₙ converges
↳ Harmonic series Σ1/n
Σaₙ converges ⇒ Σaₙ² converges
↳ aₙ = (−1)ⁿ/√n — Alternating series converges; squares give the harmonic series.
Continuous on a bounded interval ⇒ bounded
↳ f(x) = 1/x on (0,1) — Needs a compact (closed) domain.
Bounded and continuous on ℝ ⇒ uniformly continuous
↳ f(x) = sin(x²) — xₙ = √(2πn), yₙ = √(2πn + π/2) satisfy |xₙ − yₙ| → 0 while |f(xₙ) − f(yₙ)| = 1.
If lim f/g exists (0/0 form) then lim f′/g′ exists
↳ f(x) = x² sin(1/x), g(x) = x as x → 0 — f/g = x sin(1/x) → 0, but f′/g′ = 2x sin(1/x) − cos(1/x) has no limit. L'Hôpital goes one way only.
|f| Riemann integrable ⇒ f Riemann integrable
↳ f = 1 on ℚ ∩ [0,1], −1 elsewhere — |f| ≡ 1 is integrable; f is discontinuous everywhere.
If ∫₀^∞ f converges then f(x) → 0
↳ f with a spike of height n and width 2/n³ at each integer n — The total area is finite but f is unbounded, so it does not tend to 0.
fₙ → f uniformly ⇒ fₙ′ → f′
↳ fₙ(x) = sin(nx)/n — Converges uniformly to 0, derivatives cos(nx) do not converge.
Σaₙxⁿ → L as x → 1⁻ ⇒ Σaₙ = L
↳ Σ(−1)ⁿxⁿ = 1/(1 + x) → 1/2 — Σ(−1)ⁿ diverges. Abel's theorem has no converse without a Tauberian condition.
A uniformly bounded sequence in C[0,1] has a uniformly convergent subsequence
↳ fₙ(x) = xⁿ — Bounded by 1, but the pointwise limit is discontinuous so no subsequence converges uniformly. Equicontinuity is the missing hypothesis.
If all partial derivatives exist at a point then f is continuous there
↳ f(x,y) = xy/(x²+y²), f(0,0) = 0 — Both partials are 0 at the origin, but f = ½ along y = x, so f is not continuous.
A C¹ map with everywhere non-zero Jacobian is injective
↳ f(x,y) = (eˣ cos y, eˣ sin y) on ℝ² — The Jacobian determinant is e^{2x} ≠ 0, but f(x, y) = f(x, y + 2π). Invertibility is only local.
An arbitrary intersection of open sets is open
↳ ∩_{n≥1}(−1/n, 1/n) = {0} — Only finite intersections preserve openness.
Closed and bounded ⇒ compact
↳ Closed unit ball in ℓ² (or in C[0,1] with sup norm) — e₁, e₂, … has no convergent subsequence since ‖eₙ − eₘ‖ = √2. Heine–Borel is ℝⁿ-only.
Bounded ⇒ totally bounded
↳ ℝ with the discrete metric — Everything is within distance 1, but no finite set of balls of radius ½ covers it.
Completeness is a topological property
↳ (0,1) and ℝ — Homeomorphic, but ℝ is complete and (0,1) is not (1/n is Cauchy).
d(Tx, Ty) < d(x, y) for all x ≠ y on a complete space ⇒ T has a fixed point
↳ T(x) = x + 1/x on [1, ∞) — Distances strictly decrease but no contraction constant k < 1 exists; no fixed point.
Connected ⇒ path-connected
↳ Topologist's sine curve {(x, sin 1/x) : 0 < x ≤ 1} ∪ {0}×[−1,1] — Connected as the closure of a connected set; no path reaches the segment.
Closure of a path-connected set is path-connected
↳ Graph of sin(1/x) on (0,1] — Its closure is the topologist's sine curve.
A set of measure zero is countable
↳ The Cantor set — Uncountable, yet measure zero.
Pointwise convergence implies convergence of the integrals
↳ fₙ = n·1_{(0,1/n)} on [0,1] — fₙ → 0 pointwise but ∫fₙ = 1 always. Domination or monotonicity is essential.
L¹[0,1] ⊆ L²[0,1]
↳ f(x) = 1/√x — ∫f = 2 < ∞ but ∫f² = ∫dx/x = ∞. On a finite measure space the inclusion runs the other way.
An injective linear operator on a vector space is surjective
↳ The right shift on ℓ²: (x₁, x₂, …) ↦ (0, x₁, x₂, …) — Injective but misses everything with a non-zero first coordinate. Rank–nullity needs finite dimension.
Equivalent matrices are similar
↳ A = I₂ and B = 2I₂ — Both are invertible, so each is P·(the other)·Q for suitable invertible P, Q — every invertible n×n matrix is equivalent to every other, because equivalence sees nothing finer than rank. But A has eigenvalue 1 and B has eigenvalue 2, and similarity preserves eigenvalues. Equivalence allows an independent change of basis in the domain and the codomain; similarity forces the same basis on both sides.
Same characteristic polynomial ⇒ similar
↳ 0 matrix and [[0,1],[0,0]] — Both have char poly x², different minimal polynomials (x vs x²).
Same characteristic and minimal polynomial ⇒ similar
↳ 4×4 nilpotent matrices with Jordan blocks of sizes {2, 2} and {2, 1, 1} — Both have χ = x⁴ and m = x², but different numbers of blocks (ranks 2 vs 1).
AB and BA have the same minimal polynomial
↳ A = [[0,1],[0,0]], B = [[0,0],[0,1]] — AB = A has minimal polynomial x², BA = 0 has x. The characteristic polynomials do agree.
Real matrix with real eigenvalues is diagonalisable
↳ [[1,1],[0,1]] — Eigenvalue 1 with geometric multiplicity 1 < algebraic 2.
Commuting matrices are simultaneously diagonalisable
↳ A = B = [[0,1],[0,0]] — They commute but neither is diagonalisable. Need each to be diagonalisable first.
Every real matrix has a Jordan form over ℝ
↳ The rotation [[0,−1],[1,0]] — Its eigenvalues ±i are not real; over ℝ one uses the real Jordan form with a 2×2 rotation block.
A real matrix with all real eigenvalues is orthogonally diagonalisable
↳ [[1,1],[0,1]] — Eigenvalue 1 twice but only one eigenvector; orthogonal diagonalisability requires symmetry.
det A > 0 implies A is positive definite
↳ A = diag(−1, −1) — det = 1 > 0 but both eigenvalues are negative. All leading principal minors must be positive.
There exist matrices with AB − BA = I
↳ Impossible over ℝ or ℂ in finite dimensions — trace(AB − BA) = 0 but trace(I) = n ≠ 0. (It is possible for unbounded operators — the Heisenberg relation.)

24 more in this unit are part of the paid bank. Browse them.