MATH40003 SIX ANSWERS
English review edition prepared on 4 October 2026 from a preserved source copy. It is a later presentation, not the historical study interface. Source checks and difficulty judgements describe the original material author's own process; they do not indicate Imperial College London endorsement.
Source SHA-256: d53f6374cde326cd327284ba0f47684c8b4e77b024e0330018a4f7776e1f6b57Source date: 2026-08-06
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English passage 1 (11% down the source page)
Historical basis: 2026 Q1; supplementary basis: 2022 Q1 (field characteristic) and 2020 Q4 (direct sums). Gap: the earlier papers repeated this combination too often, so it is retained only once in this set. Difficulty calibration: 8 marks for routine matrix/kernel calculations, 6 for standard structural applications, 4 for proving the intersection and 2 for characteristic counterexamples; approximately 30 minutes.
English passage 2 (16% down the source page)
Recognition signal: two linear maps in opposite directions, requests for kernel/image/intersection/direct sum, and field conditions on invertibility of 2 or 3.
English passage 3 (18% down the source page)
First key step: solve the homogeneous equations for ker D, then express im G as coordinate constraints; the intersection is obtained by solving both sets of
constraints simultaneously.
English passage 4 (68% down the source page)
Marking points: 4 marks matrices; 4 kernel; 4 image/constraints; 4 zero intersection; 2 dimen-
sion/rank; 2 characteristic examples.
English passage 5 (72% down the source page)
Common errors: Only comparing dimensions without first proving the intersection is zero; dividing
by 2 or 3 without checking the field; proving only one inclusion for the image description.
English passage 6 (76% down the source page)
Final self-check: Multiply the two kernel basis vectors by D; multiply the two characteristic ex-
amples by both D and the stated G inputs; verify 2 + 3 = 5.
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English passage 1 (11% down the source page)
Historical basis: 2022 Q2 (transpose operator and change of basis) and 2021 Q3 (idempotent/nilpotent); the difficulty structure is compared with 2026 Q2, but the objects and course
method sequence come from earlier formal papers. Approximately 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: matrix spaces, transpose, symmetric/skew-symmetric subspaces, change of basis, projections and square-zero operators.
English passage 3 (17% down the source page)
First key step: find the images of the four standard basis matrices; read the later subspaces and projections directly from T (A) = ±A.
English passage 4 (72% down the source page)
Marking points: 3 linearity/matrix; 5 symmetric/skew cases; 4 basis-change calculation; 5 projec-
tions and direct sum; 3 nilpotent inverse.
English passage 5 (75% down the source page)
Common errors: Forgetting that V − changes in characteristic 2; writing the change-of-basis matrix
with rows instead of columns; asserting a projection is idempotent without expanding
T 2 = I.
English passage 6 (81% down the source page)
Final self-check: Check CE←Bej = [bj]E; check P (A) + K(A) = A; multiply (I − N )(I + N ).
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English passage 1 (11% down the source page)
Historical basis: 2026 Q3 ;supporting history: 2020 Q4(b), 2022 Q5 and 2025 Q5. This is the only
retained near-current spectral/block-structure question. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: a real symmetric positive-definite matrix followed by a complex unitary matrix written as real blocks.
English passage 3 (17% down the source page)
First key step: For part (a), exploit the invariant x-z plane and the vector e2. For part (b),
expand Z∗Z = I into real and imaginary parts before multiplying blocks.
English passage 4 (63% down the source page)
Marking points: 10 marks spectral computation/factor; 3 realification orthogonality; 3 commuta-
tion; 4 converse.
English passage 5 (67% down the source page)
Common errors: Using Q−1 without noting Q−1 = QT ; taking square roots of matrix entries instead
of eigenvalues; reversing a sign in the block transpose.
English passage 6 (70% down the source page)
Final self-check: Check all three eigenpairs; check QT Q = I; expand the four blocks of RT R.
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English passage 1 (11% down the source page)
Historical basis: 2026 Q4 ;supporting history: 2023 Q5, 2024 Q5 and 2025 Q6. This retains one
current-style group-theory ladder. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: first compute a concrete two-line permutation, then prove conjugacy properties from definitions and classify cycle types.
English passage 3 (17% down the source page)
First key step: trace each unvisited element to obtain disjoint cycles; in the general proof use the fact that conjugation simply relabels the points
in a cycle.
English passage 4 (57% down the source page)
Marking points: 8 concrete permutation marks; 3 equivalence relation; 2 cycle conjugation formula;
5 classification; 2 count.
English passage 5 (61% down the source page)
Common errors: Applying cycles left-to-right; omitting fixed points when discussing cycle type;
proving only the “conjugate implies same type”direction.
English passage 6 (64% down the source page)
Final self-check: Reconstruct the two-line map from the cycles; parity also equals (−1)10−4 where
4 is the number of cycles including the fixed point; check the count denominator.
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English passage 1 (11% down the source page)
Historical basis: 2025 Q1 (direct A coverage). The matrix size and parameter pattern are changed,
but the entry signal, case split and row/column-space methods are the official 2025 methods.
30 minutes.
English passage 2 (16% down the source page)
Recognition signal: a matrix with parameters; first REF/RREF, then rank and row/column spaces by parameter case, and finally proofs of space inclusions induced by
matrix products.
English passage 3 (20% down the source page)
First key step: Do not row-reduce immediately . First inspect pivot positions and separate
the cases b = 0 and d = 0.
English passage 4 (61% down the source page)
Marking points: 3 REF; 3 RREF; 6 rank/bases; 6 three inclusions; 2 counterexample.
English passage 5 (64% down the source page)
Common errors: Missing the non-REF case b = 0, d ̸= 0 when computing rank; using pivot columns
of a row-reduced matrix instead of the corresponding original columns; giving a coun-
terexample with incompatible sizes.
English passage 6 (69% down the source page)
Final self-check: Check every proposed row-space basis is independent; compute AB and BA
explicitly in the counterexample.
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English passage 1 (11% down the source page)
Historical basis: 2025 Q2 (all high-mark parts directly covered). Course method: subspace test, finite
basis argument, and recursive construction of an independent sequence. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: an increasing chain of subspaces, an infinite union and eventual stabilisation.
English passage 3 (17% down the source page)
First key step: For addition in the union, put both vectors into the same UN using N =
max(n1, n2). For stabilisation, use a finite basis of the union or monotone dimensions.
English passage 4 (38% down the source page)
Marking points: 2 counterexample; 5 union proof; 1 definition; 6 stabilisation; 3 polynomial chain;
3 converse.
English passage 5 (42% down the source page)
Common errors: Trying induction on the finite unions; asserting the union equals V ; using limits
or convergence of subspaces.
English passage 6 (46% down the source page)
Final self-check: In the stabilisation proof, verify every basis vector of the union lies in one common
UN; in the converse, verify each inclusion is strict.
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English passage 1 (11% down the source page)
Historical basis: 2025 Q3 direct A coverage. Course method: construct the inverse coordinate map,
transfer the vector-space axioms through a linear bijection, and compute the induced permu-
tation matrix by acting on delta functions. 30 minutes.
English passage 2 (16% down the source page)
Recognition signal: an abstract set of functions followed by an evaluation map and a permutation of the domain.
English passage 3 (18% down the source page)
First key step: Construct Φ−1 explicitly . For the operator matrix, compute T (δj) = δr−1(j), not
δr(j).
English passage 4 (58% down the source page)
Marking points: 6 vector space/isomorphism; 3 basis; 4 matrix; 3 invertibility data; 4 fixed/image.
English passage 5 (60% down the source page)
Common errors: Assuming a bijection alone creates the stated vector-space structure; using r
instead of r−1; proving only containment for the image without the dimension argument.
English passage 6 (64% down the source page)
Final self-check: Multiply the displayed matrix by each standard basis vector; check every column
sum of I −[T] is zero and its nullity is one.
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English passage 1 (11% down the source page)
Historical basis: 2025 Q4, including the 9-mark rank-by-minors proof. Course method: characteristic
polynomial/eigenspaces, then independent rows and columns to construct a non-zero minor.
30 minutes.
English passage 2 (16% down the source page)
Recognition signal: one part is a concrete eigenspace calculation, the other a proof in both directions relating all k-order minors to rank.
English passage 3 (18% down the source page)
First key step: For the theorem, prove the equivalent positive statement: rank at least k iff
some k-minor is non-zero.
English passage 4 (55% down the source page)
Marking points: 2 matrix; 8 eigenanalysis; 6 rank theorem; 4 parameter application.
English passage 5 (57% down the source page)
Common errors: Showing selected rows are independent only inside a submatrix without connect-
ing them to the original matrix; saying distinct eigenvalues imply diagonalisation but not
giving eigenspaces; at t = 2, failing to prove rank is at least two.
English passage 6 (63% down the source page)
Final self-check: Check AP = P D; verify both directions of the minor theorem; compute one
non-zero 2 × 2 minor at t = 2.
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English passage 1 (11% down the source page)
Historical basis: 2023 Q1 and 2024 Q1. Directly restores ring axioms, explicit counterexamples, and
matrix centres that were diluted in the old mocks. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: new operations or a request for the centre; return to each part of the definition.
English passage 3 (17% down the source page)
First key step: For a centre, commute a general element with matrix units. For “not a ring ”
, it is enough to exhibit one failed axiom with explicit elements.
English passage 4 (39% down the source page)
Marking points: 4 ring; 4 counterexamples; 6 centre of N3; 6 centre of full matrix ring.
English passage 5 (41% down the source page)
Common errors: Checking only commutativity of addition; giving a map that is not even a binary
operation; proving only that scalar matrices are central, not the converse.
English passage 6 (45% down the source page)
Final self-check: For the centre calculations, multiply with E12 and E23 explicitly; for Mn, verify
the scalar result works in both directions.
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English passage 1 (11% down the source page)
Historical basis: 2023 Q3. The official-answer codomain error in the old analysis is avoided: every
constructed image lies in U ∩W. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: two three-dimensional subspaces share two basis vectors, and one must construct or rule out linear maps with specified kernel/image properties.
English passage 3 (17% down the source page)
First key step: Use uniqueness of coordinates in the basis (b1, b2, u, w). Linear maps are then
defined by choosing images of these four basis vectors.
English passage 4 (46% down the source page)
Marking points: 4 intersection; 3 sum/dimension; 4 first construction; 4 impossibility; 5 second
construction.
English passage 5 (50% down the source page)
Common errors: Defining an image outside the stated codomain; checking only spanning or only
independence; forgetting that surjective endomorphisms of a finite-dimensional space are
injective.
English passage 6 (56% down the source page)
Final self-check: Write each map as a 2 × 4 coordinate matrix and verify rank/nullity directly .
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English passage 1 (11% down the source page)
Historical basis: 2024 Q3. Course method is direct bilinear expansion and standard matrix represen-
tation, not exterior algebra. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal :
“bilinear + beta(v,v)=0”indicates expansion of β(v+w, v+w) and a skew-symmetric
matrix.
English passage 3 (18% down the source page)
First key step: Expand the alternating condition first; then use β(v, w) = vT Aw in a basis.
English passage 4 (49% down the source page)
Marking points: 4 skew result; 5 matrix test; 5 rank in R3; 6 classification in R2.
English passage 5 (51% down the source page)
Common errors: Using 2−1 without stating the field; saying an odd skew matrix has zero deter-
minant without explaining why or giving the rank lower bound; confusing symmetric and
alternating.
English passage 6 (56% down the source page)
Final self-check: Check AT = −A entry by entry and calculate the displayed 2 × 2 minors.
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English passage 1 (11% down the source page)
Historical basis: 2023 Q5-Q6 and 2024 Q5-Q6. It combines dihedral relations, automorphism groups,
inner automorphisms and commutators, but each part uses the official direct group method.
30 minutes.
English passage 2 (16% down the source page)
Recognition signal: rks, conjugation maps, Aut(G), and the commutator subgroup.
English passage 3 (18% down the source page)
First key step: Establish sr = r−1s first. It reduces every product/inverse in D to one of two
normal forms.
English passage 4 (58% down the source page)
Marking points: 7 dihedral subgroup; 3 Aut group; 5 inner homomorphism/kernel; 3 normality; 2
abelian quotient.
English passage 5 (62% down the source page)
Common errors: Assuming r and s commute; checking closure only for one of the four product
types; proving properties only for a single commutator rather than products of them.
English passage 6 (65% down the source page)
Final self-check: Reduce srk to r−ks; check Cgh(x) on an arbitrary x; verify the quotient equality
with the coset criterion.
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English passage 1 (11% down the source page)
Historical basis :2020 Q2, 2021 Q1 and 2022 Q1. It directly trains one row reduction across different
fields and the affine-coset theorem. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: solve the same system over several fields, then explain the general solution-set structure .
English passage 3 (17% down the source page)
first key step :Perform row operations over the integers as far as possible before reducing
modulo the field characteristic.
English passage 4 (46% down the source page)
Marking points :8 field solutions; 6 fibre/coset theorem; 6 rank criterion and dimension.
English passage 5 (48% down the source page)
Common errors :Dividing by 6 in F2 or F3; treating a finite-field free parameter as infinitely many
real values; asserting the coset form without first choosing a particular solution.
English passage 6 (52% down the source page)
Final self-check :Substitute each solution family into the original equations in the appropriate
field; verify the affine dimension equals the number of free variables.
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English passage 1 (11% down the source page)
Historical basis :2021 Q3 and 2022 Q3. The main methods are direct linearity/idempotence checks,
image-kernel decomposition, the independent-chain proof for nilpotent degree, and finite geo-
metric series. 30 minutes.
English passage 2 (16% down the source page)
Recognition signal: the appearance of T 2 = T or Nm = 0.
English passage 3 (18% down the source page)
first key step :For idempotents, write v = Tv + (v −Tv). For nilpotents, use v, Nv, . . . , N m−1v
and apply powers of N successively.
English passage 4 (49% down the source page)
Marking points :6 classification; 5 idempotent decomposition; 5 nilpotent degree; 4 explicit in-
verses.
English passage 5 (53% down the source page)
Common errors :Calling a non-linear map idempotent under the course definition; claiming im T =
ker T for a non-zero idempotent; applying only one power of N and concluding all coeffi-
cients vanish.
English passage 6 (58% down the source page)
Final self-check :Check every map sends 0 correctly; multiply the finite series; test the shift on bn.
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English passage 1 (11% down the source page)
Historical basis :2021 Q4. Course method: diagonalise the 2 × 2 recurrence matrix and use the or-
thonormal eigenbasis of a symmetric matrix for norm estimates. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: a two-dimensional recurrence plus a general real symmetric norm statement.
English passage 3 (17% down the source page)
first key step :Write (an, bn)T = Mn−1(a1, b1)T . For the norm theorem, expand an arbitrary
unit vector in an orthonormal eigenbasis.
English passage 4 (59% down the source page)
Marking points :10 recurrence; 2 Pythagoras; 8 norm extrema.
English passage 5 (61% down the source page)
Common errors :Using Mn instead of Mn−1; forgetting to normalise eigenvectors in the spectral
theorem; proving only inequalities and not attainment.
English passage 6 (65% down the source page)
Final self-check :Check the closed formula at n = 1; run one recurrence step; plug a unit eigen-
vector into the extrema formula.
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English passage 1 (11% down the source page)
Historical basis :2020 Q6 and 2024 Q6(c). The methods are cyclic-group arithmetic, Lagrange, the
kernel/coset criterion, and standard explicit homomorphisms. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: finite cyclic groups, element orders, coprime-order subgroups, homomorphism kernels and cosets .
English passage 3 (17% down the source page)
first key step :Use ord(gk) = 12/ gcd(12, k), and for the fibre statement rewrite equality as
x−1y ∈ker ϕ.
English passage 4 (53% down the source page)
Marking points :5 order distribution; 4 subgroup product; 5 kernel/fibres; 3 prime-order injection;
3 examples.
English passage 5 (57% down the source page)
Common errors :Listing subsets that are not subgroups; assuming AB = G from orders without
proving injectivity or using the product formula; using inverse or transpose as a homomor-
phism on GL2 without checking order reversal.
English passage 6 (62% down the source page)
Final self-check :Sum the element counts to 12; check the intersection order; multiply the shear
matrices explicitly.
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English passage 1 (11% down the source page)
Historical basis: 2021 Q5, 2020 Q5 and 2025 Q4(b). It directly covers the previously under-trained
high-mark determinant/minor ability chain. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: sparse matrices, eigenspace dimensions and the largest nonzero minor.
English passage 3 (17% down the source page)
First key step: For the monomial matrix, permute rows to a diagonal matrix. For rank, prove
“rank at least k iff some k-minor is non-zero”.
English passage 4 (46% down the source page)
Marking points: 4 monomial determinant; 4 added entry; 5 characteristic factor; 7 rank theorem.
English passage 5 (49% down the source page)
Common errors: Expanding a huge determinant instead of using the structure; assuming geomet-
ric multiplicity automatically implies algebraic multiplicity without the basis argument;
proving only one direction of the rank theorem.
English passage 6 (54% down the source page)
Final self-check :Test part (b) on a 2 × 2 example; verify the block matrix has µI2 in the first two
columns; identify the selected rows and columns in the rank proof.
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English passage 1 (11% down the source page)
Historical basis: 2024 Q4(b). Directly restores the tridiagonal determinant recurrence and the 2x2
diagonalisation method. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: a tridiagonal matrix determinant dn and request for a closed form.
English passage 3 (17% down the source page)
First key step: Expand along the last row, then the last column of the remaining minor, to
obtain a second-order recurrence.
English passage 4 (52% down the source page)
Marking points: 5 recurrence; 3 matrix form; 8 diagonalisation/closed form; 4 numerical check.
English passage 5 (54% down the source page)
Common errors :Using b + c as both the diagonal and an eigenvalue; omitting d0 = 1; shifting the
power by one.
English passage 6 (58% down the source page)
Final self-check: Check n = 0 and n = 1 in the closed formula; substitute it into the recurrence
algebraically.
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English passage 1 (11% down the source page)
Historical basis: 2025 Q5 direct A/B coverage and 2020 Q1(f). It combines the 2025 QR/integers
question with a concise O2 classification, while retaining a two-hour workload. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: apply Gram-Schmidt to three given columns, then classify a matrix group using the integer constraints on orthogonal columns.
English passage 3 (17% down the source page)
first key step :Compute q1, subtract its projection from v2, then subtract both projections
from v3; the R entries are qi · vj.
English passage 4 (59% down the source page)
Marking points: 8 QR; 5 O2 classification; 7 signed permutation classification/group/order.
English passage 5 (61% down the source page)
Common errors: Forgetting to square the norm in projection denominators; allowing a zero or
two non-zero integer entries in a unit column; proving only that permutation matrices are
included, not classifying all of On(Z).
English passage 6 (66% down the source page)
Final self-check: Check QT Q = I and QR = A; count choices column by column; verify the two O2
forms have determinant ±1.
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English passage 1 (11% down the source page)
Historical basis: 2021 Q6, 2023 Q4(d), 2022 Q6(d) and the finite-group counting result. This gap paper
restores several rotation abilities but keeps each proof short and independent. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: the set of squares, power maps on roots of unity, permutation cycle types, counting elements of prime order, and the Cayley action.
English passage 3 (17% down the source page)
First key step: Use the subgroup test for squares in an abelian group; use element orders
for the power map; count order-p elements by their unique cyclic subgroups.
English passage 4 (49% down the source page)
Marking points: 6 squares; 4 unit roots; 4 cycle count; 3 divisibility; 3 Cayley.
English passage 5 (51% down the source page)
Common errors: Assuming (gh)2 = g2h2 in a non-abelian group; failing to divide by 2! when counting
two 3-cycles; not proving distinct order-p subgroups meet trivially.
English passage 6 (55% down the source page)
Final self-check: For the power map, check kernel via orders; sum the two cycle-shape counts;
evaluate the Cayley embedding at the identity.
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English passage 1 (11% down the source page)
Historical basis: 2025 Q3, 2026 Q2 and 2022 field-characteristic work. It changes the wording and
cycle lengths to test entry recognition, not memory of the current paper. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: a function space plus a permutation with cycles of lengths 2,3,1; ask fixed space, differ-
ence image and characteristic exceptions.
English passage 3 (19% down the source page)
First key step: Use delta coordinates. Analyse I − T separately on each cycle of τ .
English passage 4 (47% down the source page)
Marking points: 4 isomorphism/basis; 4 operator matrix; 4 fixed space; 4 image/rank; 4 character-
istic intersection.
English passage 5 (51% down the source page)
Common errors: Using τ instead of τ −1 in the matrix; assuming the fixed space has dimension
one; dividing by the cycle length without checking characteristic.
English passage 6 (55% down the source page)
Final self-check: Check one basis delta function from each cycle; verify the sum constraints;
substitute the characteristic intersection vectors.
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English passage 1 (11% down the source page)
Historical basis: 2020 Q4(a), 2021 idempotents and 2026 characteristic exceptions. The polynomial
has three distinct roots only when 2 is invertible. 30 minutes.
English passage 2 (14% down the source page)
Recognition signal: an operator satisfying A3 = A and the field condition that 2 is invertible.
English passage 3 (17% down the source page)
First key step: Factor x3 − x = x(x − 1)(x + 1) and construct the three Lagrange interpolation
projectors.
English passage 4 (53% down the source page)
Marking points: 4 A2 facts; 10 projectors; 3 direct sum/diagonalisation; 3 characteristic-two exam-
ple.
English passage 5 (57% down the source page)
Common errors: Using the projectors when 2 is not invertible; proving the images are con-
tained in eigenspaces but not the reverse; claiming repeated eigenvalue alone means
non-diagonalisable.
English passage 6 (62% down the source page)
Final self-check: Expand every projector polynomial using A3 = A; apply P+ to a vector with Av = v;
square the characteristic-two matrix.
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English passage 1 (11% down the source page)
Historical basis: 2026 Q3(a), 2021 Q4(b), 2022 Q4(c) and 2025 Q5(a). It mixes three historical entry
signals while preserving the 10/10 computation-proof balance. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: symmetric matrix followed by unit-sphere norm and a general QR existence
proof.
English passage 3 (19% down the source page)
First key step: Use the evident eigenvector (1, 1, 1) and its orthogonal complement; for QR,
apply Gram-Schmidt to the columns.
English passage 4 (51% down the source page)
Marking points: 6 diagonalisation; 3 square-root factor; 5 norm theorem; 6 QR existence.
English passage 5 (54% down the source page)
Common errors: Using non-orthogonal eigenvectors in Q; confusing the matrix square root with
entrywise square roots; failing to explain why R is upper triangular.
English passage 6 (57% down the source page)
Final self-check: Check ui·uj = δij; check eigenpairs; verify each column of QR is the corresponding
original column.
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English passage 1 (10% down the source page)
Historical basis: 2025 Q6 direct A/B coverage, with the added Z2 entry from 2023 Q6. It is the final
transfer test for reading group descriptions. 30 minutes.
English passage 2 (15% down the source page)
Recognition signal: concrete permutation products, maximum order, cyclicity of four groups, and a simplified subgroup test under a finite-order assumption.
English passage 3 (17% down the source page)
First key step: For the product, apply the rightmost cycle first. For cyclicity , use a structural
invariant: commutativity, rank, or an explicit generator.
English passage 4 (43% down the source page)
Marking points: 4 product/order/sign; 4 maximum order; 5 cyclicity; 7 subgroup theorem.
English passage 5 (45% down the source page)
Common errors: Applying permutation factors left-to-right; saying S5 is not cyclic only because
it lacks an element of order 120 without justifying possible orders; forgetting integer
negative powers in the shear group.
English passage 6 (51% down the source page)
Final self-check: Reconstruct the product on all six points; check the least common multiple for
the relevant partitions; multiply two shear matrices.