02 MATH40003 SEVEN YEAR ABILITY UNITS
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: b34f0b18a98925fd75b98c41c8be61333470ac3dde8473c5e240092898ae3debSource date: 2026-08-06
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Complete English table: 194 historical ability units
Reconstructed from the preserved source CSV, matched to the PDF by unit ID, year, question, marks and coverage grades. The mathematical notation is retained. Some source descriptions and first steps were already truncated; these are explicitly marked rather than reconstructed as new evidence.
| Unit | Year | Question | Marks | Codes | Ability / question type | First key step | Time | Workload (as in PDF) | Old | New |
|---|---|---|---|---|---|---|---|---|---|---|
| AU001 | 2020 | Q1(a) | 2 | M02,M05 | Find the inverse of the given 3-by-3 matrix by elementary row operations. / Calculation/solution | Reduce (A|I) to (I|A^{-1}). | 3.0 min | Low / Medium/low | C | B |
| AU002 | 2020 | Q1(b)-(c) | 2 | S05,M05 | Define orthogonal matrices and verify A^{-1}=A^T. / Definition/structure | Compare the inverse with the transpose. | 3.0 min | Low / Medium/low | A | A |
| AU003 | 2020 | Q1(d) | 4 | L02,S05 | Derive the two-dimensional rotation matrix from images of basis vectors. / Definition/structure | Calculate R_theta(e1), R_theta(e2). | 6.0 min | Medium / Medium/low | B | A |
| AU004 | 2020 | Q1(e) | 3 | S05 | Write two reflection matrices and prove their composition is a rotation. / Proof | Substitute into the reflection formula and multiply. | 4.5 min | Low / Medium/low | B | A |
| AU005 | 2020 | Q1(f) | 7 | S05,M06 | Classify all real 2-by-2 orthogonal matrices as rotations or reflections. / Definition/structure | Use det=±1 and orthogonal columns to derive the standard form. | 10.5 min | High / Medium-high | A | A |
| AU006 | 2020 | Q1(g) | 2 | L02,S05 | Find the matrix of a three-dimensional rotation about a given axis in an orthonormal basis. / Definition/structure | Fix the axis direction and use a 2D rotation in the perpendicular plane. | 3.0 min | Low / Medium-high | B | A |
| AU007 | 2020 | Q2(a) | 1 | M04 | Write the augmented matrix of a linear system. / Calculation/solution | Arrange coefficients in variable order. | 1.5 min | Low / Medium/low | C | A |
| AU008 | 2020 | Q2(b) | 6 | M03,M04 | Classify existence of solutions by a,b and find all solutions. / Calculation/solution | Perform row reduction, then identify contradictions and free variables. | 9.0 min | Medium / Medium/low | A | A |
| AU009 | 2020 | Q2(c) | 5 | M07,M04 | Define row space/rank and find them for A and (A|c). / Calculation/solution | Nonzero rows after row reduction form a row-space basis. | 7.5 min | Medium / Medium-high | B | B |
| AU010 | 2020 | Q2(d) | 8 | V09,L03,L04,M04 | Prove the solution set of a consistent system is a coset of a subspace of dimension n-rank(A). / Proof | Take a particular solution c1 and prove S=c1+ker A. | 12.0 min | High / High | A | A |
| AU011 | 2020 | Q3(a) | 3 | V02 | State the subspace criterion. / Definition/structure | Nonempty, closed under addition and scalar multiplication. | 4.5 min | Low / Medium/low | D | A |
| AU012 | 2020 | Q3(b) | 4 | V02,V04,V08 | Prove polynomials whose coefficients satisfy the recurrence form a subspace and find a basis. / Proof | The coefficients are uniquely determined by a0. | 6.0 min | Medium-high / Medium/low | C | B |
| AU013 | 2020 | Q3(c) | 8 | L01,L03,V02 | Define linear maps, kernels and images, and prove the kernel is a subspace. / Proof | Apply linearity term by term. | 12.0 min | High / Medium/low | B | B |
| AU014 | 2020 | Q3(d) | 5 | L01,L03,L10,V03 | Verify linearity of T_n and find image and kernel; then assess U_n [source description ends here]. / True-or-false/counterexample | Describe image and kernel using a0, then check the set [source wording incomplete]. | 7.5 min | Medium / High | B | B |
| AU015 | 2020 | Q4(a)(i) | 3 | M11,M05 | From A^2-3A+2I=0, prove A is invertible and find its inverse. / Proof | Construct the two-sided inverse (3I-A)/2. | 4.5 min | Low / Medium-high | A | A |
| AU016 | 2020 | Q4(a)(ii)-(iii) | 5 | S01,S02,V03 | Prove E1 and E2 sum directly to the whole space and deduce diagonalisability. / Proof | Write any v as (A-I)v-(A-2I)v. | 7.5 min | Medium-high / Medium/low | A | A |
| AU017 | 2020 | Q4(a)(iv) | 2 | S01,L10 | Deduce v belongs to E1 from boundedness of A^k v. / Definition/structure | Write v=v1+v2 and eliminate the 2^k component. | 3.0 min | Low / Medium-high | C | B |
| AU018 | 2020 | Q4(a)(v) | 2 | S01,S02,M10 | List the three similarity normal forms in dimension 2. / Definition/structure | I, 2I, diag(1,2). | 3.0 min | Low / Medium/low | D | A |
| AU019 | 2020 | Q4(b)(i) | 2 | S07 | Prove B^TB is positive definite. / Proof | v^TB^TBv=||Bv||^2. | 3.0 min | Low / Medium/low | B | A |
| AU020 | 2020 | Q4(b)(ii) | 6 | S03,S08,S07 | Derive an SVD-type decomposition B=QDP for an invertible matrix using the spectral theorem. / Definition/structure | Orthogonally diagonalise B^TB and construct Q. | 9.0 min | Medium / High | A | A |
| AU021 | 2020 | Q5(a) | 6 | M06 | Define the determinant recursively and prove scaling one row scales it by the same factor. / Proof | Expand along the first row and induct on matrix order. | 9.0 min | Medium-high / Medium/low | A | A |
| AU022 | 2020 | Q5(b) | 3 | M06 | Calculate a composite matrix determinant using multiplicativity and transpose properties. / Calculation/solution | Split it into powers of the factors' determinants. | 4.5 min | Low / Medium/low | B | A |
| AU023 | 2020 | Q5(c)(i)-(iii) | 7 | G07 | Define left cosets and prove cosets of one subgroup are equal or disjoint [source continues incompletely]. / Proof | Take an intersection point k and write X=kH; construct by left multiplication. | 10.5 min | High / Medium/low | B | B |
| AU024 | 2020 | Q5(c)(iv) | 4 | G07,G11 | Characterise intersections of cosets of different subgroups and give two kinds of example. / Definition/structure | If the intersection is nonempty, take g in it [source proof instruction ends here]. | 6.0 min | Medium / Medium-high | B | B |
| AU025 | 2020 | Q6(a)(i)-(iii) | 6 | G02,G03 | Define element order and count elements of each order in a cyclic group of order 12. / Definition/structure | ord(g^k)=12/gcd(k,12). | 9.0 min | Medium / Medium/low | A | A |
| AU026 | 2020 | Q6(a)(iv) | 3 | G07,G11 | Find proper subgroups A,B with trivial intersection and AB=G. / Definition/structure | Use subgroups of orders 3 and 4 and Lagrange's theorem. | 4.5 min | Low / High | C | A |
| AU027 | 2020 | Q6(b) | 6 | G05,G06,G01 | Define homomorphism and kernel; prove the kernel is a subgroup and equal images occur precisely in one coset. / Proof | Use the subgroup test and g1^{-1}g2∈kerφ. | 9.0 min | Medium-high / Medium-high | A | A |
| AU028 | 2020 | Q6(c) | 5 | G05,G13 | Construct four types of nontrivial homomorphism. / Construction | Conjugation, exponent, determinant and a unit upper-triangular representation [last phrase truncated in source]. | 7.5 min | Medium-high / High | B | A |
| AU029 | 2021 | Q1(a) | 10 | M03,M04 | Classify a linear system by alpha and give the unique or infinitely many solutions. / Definition/structure | Retain parameter-dependent pivots and identify exceptional values. | 15.0 min | High / Medium/low | A | A |
| AU030 | 2021 | Q1(b) | 2 | V02 | Use only vector-space axioms to prove a set without zero is not a subspace. / Proof | Prove every nonempty subspace contains 0. | 3.0 min | Low / Medium/low | D | C |
| AU031 | 2021 | Q1(c)(i) | 2 | V02,V09 | Determine whether the original nonhomogeneous solution set S is a subspace. / True-or-false/counterexample | A nonhomogeneous solution set generally does not contain 0. | 3.0 min | Low / Medium/low | B | A |
| AU032 | 2021 | Q1(c)(ii) | 3 | V03,V09 | Analyse a specified translation of the solution set and determine whether it is a subspace. / True-or-false/counterexample | Separate exceptional and ordinary alpha values. | 4.5 min | Low / Medium-high | C | A |
| AU033 | 2021 | Q1(c)(iii) | 3 | V03,V09 | Analyse another translation of the solution set and determine whether it is a subspace. / True-or-false/counterexample | Check whether the translation passes through the origin. | 4.5 min | Low / Medium-high | C | A |
| AU034 | 2021 | Q2(a) | 3 | V02,V08 | Prove the parametrised set W is a subspace of R4. / Proof | Add and scale the parameters. | 4.5 min | Low / Medium/low | C | A |
| AU035 | 2021 | Q2(b) | 5 | V04,V05 | Prove four given vectors form a basis and find the B-coordinates of a general vector. / Proof | Prove spanning/independence and solve for coordinates. | 7.5 min | Medium-high / Medium-high | B | B |
| AU036 | 2021 | Q2(c) | 5 | L02,L03,L04,V05 | Find a linear map's image, image dimension and matrix in nonstandard bases. / Calculation/solution | Express images of basis vectors in the codomain basis. | 7.5 min | Medium / Medium/low | B | B |
| AU037 | 2021 | Q2(d)(i) | 4 | L01 | Prove the composition operator T*:f↦T∘f is linear. / Proof | Compare the functions pointwise. | 6.0 min | Medium-high / High | B | B |
| AU038 | 2021 | Q2(d)(ii) | 3 | L04,L06 | Find the rank of the matrix of T*. / Calculation/solution | Left-multiply by a fixed matrix and calculate the image dimension. | 4.5 min | Low / High | B | B |
| AU039 | 2021 | Q3(a) | 11 | L01,L07 | Determine whether five maps are linear and idempotent. / True-or-false/counterexample | Check linearity and T^2=T separately. | 16.5 min | High / Medium-high | A | A |
| AU040 | 2021 | Q3(b) | 2 | L01 | Prove a composite of linear maps is linear. / Proof | Use linearity of T, then of S. | 3.0 min | Low / Medium/low | C | A |
| AU041 | 2021 | Q3(c) | 4 | L07,L09 | Determine whether a composite of two idempotents must be idempotent and give a counterexample. / True-or-false/counterexample | Construct noncommuting projections. | 6.0 min | Medium / High | B | A |
| AU042 | 2021 | Q3(d) | 3 | L07,V03 | Discuss whether an idempotent T can satisfy ImT=kerT. / Definition/structure | Correct conclusion: impossible for nonzero V [the source's following official-solution note is truncated]. | 4.5 min | Low / Medium/low | B | A |
| AU043 | 2021 | Q4(a)(i) | 6 | S01,S09 | Write a two-dimensional recurrence as a matrix and find eigenvalues/vectors. / Calculation/solution | Use the characteristic polynomial and eigenvectors. | 9.0 min | Medium / Medium/low | A | A |
| AU044 | 2021 | Q4(a)(ii)-(iii) | 4 | S09,S02 | Find the general term and recover initial values from later values. / Calculation/solution | Diagonalise or observe A^2=I. | 6.0 min | Medium / Medium/low | A | A |
| AU045 | 2021 | Q4(b)(i) | 2 | S04 | Prove the squared-norm formula for a linear combination of orthogonal vectors. / Proof | Orthogonality eliminates cross terms. | 3.0 min | Low / Medium/low | C | A |
| AU046 | 2021 | Q4(b)(ii) | 8 | S03,S10,S04 | Find the maximum and minimum of ||Av|| on the unit sphere for a real symmetric matrix. / Calculation/solution | Expand in an orthonormal eigenbasis. | 12.0 min | High / High | B | A |
| AU047 | 2021 | Q5(a)(i) | 3 | M06 | Prove the determinant of a matrix with one nonzero entry per row and column is ± a nonzero product [source wording truncated]. / Proof | Reduce it to diagonal form. | 4.5 min | Low / Medium/low | A | A |
| AU048 | 2021 | Q5(a)(ii) | 3 | M06 | Explain why replacing one zero entry by 1 leaves the determinant unchanged. / Definition/structure | Use the row with the existing nonzero entry to eliminate the new 1. | 4.5 min | Low / Medium/low | A | A |
| AU049 | 2021 | Q5(b) | 6 | S01,M08 | Use a two-dimensional eigenspace to prove a characteristic-polynomial factor [source does not state the factor]. / Proof | Extend the eigenvector basis and expand the determinant. | 9.0 min | Medium-high / High | A | A |
| AU050 | 2021 | Q5(c)(i) | 3 | G01,G07 | Prove the stabiliser H is a subgroup of S6 and find its order. / Proof | Permute {1,2} and the other four points separately. | 4.5 min | Low / Medium/low | D | B |
| AU051 | 2021 | Q5(c)(ii)-(iii) | 5 | G07 | Characterise cosets by the unordered images g(1),g(2), and find the index in two ways. / Calculation/solution | Algebraic proof and combinatorial counting. | 7.5 min | Medium / Medium-high | B | B |
| AU052 | 2021 | Q6(a) | 8 | P06,G01 | Determine whether squares form a subgroup or equal a specified set in four group settings [source ending truncated]. / True-or-false/counterexample | Abelian groups, odd order, sign in S_n and dihedral groups. | 12.0 min | High / Medium-high | A | A |
| AU053 | 2021 | Q6(b)(i) | 4 | P01,P02 | Compute disjoint cycles and orders of several composed permutations in S9. / Calculation/solution | Trace cycles and take the least common multiple. | 6.0 min | Medium / Medium/low | A | A |
| AU054 | 2021 | Q6(b)(ii) | 4 | P02 | List cycle types and count elements of order 3 in S9. / Definition/structure | Choose supports and divide out overcounting. | 6.0 min | Medium / Medium/low | B | A |
| AU055 | 2021 | Q6(c) | 4 | G03,P07 | Prove the number of elements of order p in a finite group is divisible by p-1. / Proof | Each subgroup of order p has p-1 generators. | 6.0 min | Medium-high / High | C | A |
| AU056 | 2022 | Q1(a)(i)-(iii) | 8 | M03,L12 | Solve the same linear system over R, F5 and F7. / Calculation/solution | Perform integer row reduction once, then interpret separately in each field [source qualification truncated]. | 12.0 min | High / Medium/low | C | A |
| AU057 | 2022 | Q1(b) | 4 | V01 | Prove additive cancellation from vector-space axioms. / Proof | Use commutativity, additive inverses, associativity and zero. | 6.0 min | Medium-high / Medium/low | D | C |
| AU058 | 2022 | Q1(c)(i) | 4 | V01 | Verify specified vector-space axioms for V×W. / Definition/structure | Reduce the axioms to those of V and W. | 6.0 min | Medium / Medium/low | D | C |
| AU059 | 2022 | Q1(c)(ii) | 4 | V01,V02 | Prove A×B is a subspace of V×W. / Proof | Check closure coordinate by coordinate. | 6.0 min | Medium-high / Medium/low | D | C |
| AU060 | 2022 | Q2(a) | 4 | V04,V08 | Prove four given 2-by-2 matrices form a basis of M2(R). / Proof | Express the standard basis in them, or calculate a determinant. | 6.0 min | Medium-high / Medium/low | C | A |
| AU061 | 2022 | Q2(b)(i)-(ii) | 4 | L02,V05 | Find the transpose operator's matrix in two bases. / Calculation/solution | Transpose each basis vector and compute its coordinates. | 6.0 min | Medium / Medium/low | A | A |
| AU062 | 2022 | Q2(b)(iii) | 3 | V05,M10 | Find the identity map's change-of-basis matrix. / Calculation/solution | Express new basis vectors in old-basis coordinates. | 4.5 min | Low / Medium/low | A | A |
| AU063 | 2022 | Q2(c) | 7 | L02,L01 | Prove that a map with matrix I in a basis is the identity. / Proof | It fixes all basis vectors; apply linearity. | 10.5 min | High / Medium-high | B | C |
| AU064 | 2022 | Q2(d) | 2 | L03,L04 | Find the rank and nullity of a linear functional. / Calculation/solution | Construct a preimage of any real number. | 3.0 min | Low / Medium/low | B | B |
| AU065 | 2022 | Q3(a) | 2 | L08,L02 | Prove the shift operator is nilpotent. / Proof | T^n=0. | 3.0 min | Low / Medium/low | A | A |
| AU066 | 2022 | Q3(b) | 4 | L08,V02 | Prove Nil(T) is a subspace. / Proof | For u,v choose a sufficiently large common power. | 6.0 min | Medium-high / Medium/low | C | A |
| AU067 | 2022 | Q3(c) | 8 | L08,V06 | Prove the nilpotency index does not exceed dimV. / Proof | Correct completion: apply T successively to eliminate coefficients. | 12.0 min | High / High | A | A |
| AU068 | 2022 | Q3(d) | 6 | L09 | Prove I-T and I+T are invertible. / Proof | Write explicit inverse polynomials. | 9.0 min | Medium-high / Medium/low | A | A |
| AU069 | 2022 | Q4(a)(i) | 2 | L02,V05 | Write the matrix from images of basis vectors in a nonstandard basis. / Calculation/solution | Each column consists of the B-coordinates of T(v_i). | 3.0 min | Low / Medium/low | B | B |
| AU070 | 2022 | Q4(a)(ii) | 6 | S01,S02 | Find eigenspaces and use insufficient geometric multiplicity to rule out diagonalisation. / True-or-false/counterexample | Find the dimension of the repeated eigenvalue's eigenspace. | 9.0 min | Medium / Medium/low | C | B |
| AU071 | 2022 | Q4(b)(i) | 3 | M06,M05 | Find a parameter matrix's determinant and determine invertibility. / True-or-false/counterexample | det=yz(x+1) [the source's note about an official error involving -1 is truncated]. | 4.5 min | Low / Medium/low | B | B |
| AU072 | 2022 | Q4(b)(ii) | 2 | M10,M06 | Rule out P using equality of determinants of similar matrices. / Definition/structure | Compare detB and detA. | 3.0 min | Low / Medium/low | B | B |
| AU073 | 2022 | Q4(c) | 7 | S04,S06 | Prove every invertible real matrix has a QR decomposition. / Proof | Orthogonalise its columns and read the upper-triangular coefficients. | 10.5 min | High / High | A | A |
| AU074 | 2022 | Q5(a)(i) | 3 | S04 | Check three vectors form an orthogonal basis. / Definition/structure | Pairwise dot products are 0 and each vector is nonzero. | 4.5 min | Low / Medium/low | C | A |
| AU075 | 2022 | Q5(a)(ii) | 3 | S03 | Deduce matrix symmetry from T(u)·v=u·T(v). / Proof | u^T A^T v=u^TAv for all u,v. | 4.5 min | Low / Medium/low | C | A |
| AU076 | 2022 | Q5(a)(iii) | 2 | S03,S01 | Use the spectral theorem to prove a symmetric matrix's characteristic polynomial splits over R. / Proof | It is orthogonally similar to a real diagonal matrix. | 3.0 min | Low / Medium/low | B | A |
| AU077 | 2022 | Q5(b) | 3 | S05 | Prove the two-dimensional rotation matrix is orthogonal. / Proof | Calculate AA^T=I directly. | 4.5 min | Low / Medium/low | B | A |
| AU078 | 2022 | Q5(c)(i) | 2 | S12 | Prove permutation matrices are orthogonal. / Proof | The columns permute the standard basis. | 3.0 min | Low / Medium/low | B | A |
| AU079 | 2022 | Q5(c)(ii) | 7 | S12,G01,G04,P01 | Prove permutation matrices form a subgroup isomorphic to S_n. / Proof | Use sigma↦P_sigma and verify multiplication [source continues incompletely]. | 10.5 min | High / High | A | A |
| AU080 | 2022 | Q6(a)(i) | 2 | G01 | Prove a line in R2 is an additive subgroup. / Proof | Add and negate its parameter. | 3.0 min | Low / Medium/low | D | C |
| AU081 | 2022 | Q6(a)(iii) | 2 | G04 | Construct an isomorphism of this subgroup with (R,+). / Construction | (x,2x)↦x. | 3.0 min | Low / Medium/low | D | C |
| AU082 | 2022 | Q6(a)(iv) | 4 | G08,G07 | Define an operation on cosets and prove the quotient group is isomorphic to R. / Proof | Use the linear functional b-2a for a simpler interpretation. | 6.0 min | Medium-high / Medium/low | D | B |
| AU083 | 2022 | Q6(b) | 3 | P01 | Convert two-line notation to disjoint cycles. / Definition/structure | Start from each unvisited element. | 4.5 min | Low / Medium/low | B | B |
| AU084 | 2022 | Q6(c)(i)-(ii) | 5 | G08,G07,G05 | Prove index-2 subgroups are normal and apply to A5=ker(sgn). / Proof | H and its complement are the only two cosets. | 7.5 min | Medium-high / Medium-high | B | B |
| AU085 | 2022 | Q6(d) | 4 | G10 | Prove a finite group embeds in some S_n. / Proof | g↦(h↦gh) is an injective homomorphism. | 6.0 min | Medium-high / High | A | A |
| AU086 | 2023 | Q1(a) | 3 | M05 | Use repeated rows to show a matrix is not invertible. / True-or-false/counterexample | Row reduction produces a zero row. | 4.5 min | Low / Medium/low | D | B |
| AU087 | 2023 | Q1(b) | 4 | M01 | Prove matrix-multiplication distributivity from definitions. / Proof | Compare (i,j) entries. | 6.0 min | Medium-high / Medium/low | D | A |
| AU088 | 2023 | Q1(c) | 6 | M13,M01 | Prove M_n(R) is a ring under usual operations. / Proof | Check additive abelian-group structure, associative multiplication and both distributive laws. | 9.0 min | Medium-high / Medium/low | A | A |
| AU089 | 2023 | Q1(d)(i)-(ii) | 4 | M13 | Show two custom operations fail to form rings and give counterexamples. / True-or-false/counterexample | Identity, associativity or distributivity fails. | 6.0 min | Medium / High | A | A |
| AU090 | 2023 | Q1(e) | 3 | M13 | Construct a binary operation failing both distributive laws. / Construction | Ensure closure first, then give explicit triples. | 4.5 min | Low / High | C | C |
| AU091 | 2023 | Q2(a) | 6 | M05,M03 | Find invertibility conditions and the inverse of a parameter-dependent 3-by-3 matrix. / Definition/structure | Exclude two parameter values. | 9.0 min | Medium / Medium/low | C | C |
| AU092 | 2023 | Q2(b) | 6 | M03 | Translate eye-colour and teacher/student counts into a five-variable system and solve it. / Calculation/solution | Define variables, write equations and row-reduce. | 9.0 min | Medium / Medium/low | C | C |
| AU093 | 2023 | Q2(c)(i) | 2 | G03 | Find finite-order elements in the additive group of a real vector space. / Calculation/solution | nv=0 and characteristic 0 imply v=0. | 3.0 min | Low / Medium/low | D | C |
| AU094 | 2023 | Q2(c)(ii) | 3 | G01,V02 | Disprove that every additive subgroup is a subspace. / True-or-false/counterexample | Z≤(R,+) is not closed under scalar multiplication by 1/2. | 4.5 min | Low / Medium-high | D | C |
| AU095 | 2023 | Q2(d) | 3 | G01,V03 | Prove the union of two subgroups, neither containing the other, is not a subgroup. / Proof | Choose h∈H\K and k∈K\H; examine hk. | 4.5 min | Low / Medium/low | C | C |
| AU096 | 2023 | Q3(a) | 3 | V02,V03 | Prove U∩W is a subspace. / Proof | Apply the subspace test. | 4.5 min | Low / Medium/low | B | A |
| AU097 | 2023 | Q3(b) | 3 | V04,V06 | Prove two given vectors form a basis of U∩W. / Proof | First find dimension 2, then prove independence. | 4.5 min | Low / Medium/low | B | A |
| AU098 | 2023 | Q3(c) | 6 | V06,V04 | Extend a basis of the intersection to bases of U and W. / Definition/structure | Choose vectors outside the intersection and prove independence. | 9.0 min | Medium / Medium/low | B | A |
| AU099 | 2023 | Q3(d) | 2 | V03,V06 | Prove any chosen complementary vectors together form a basis of V. / Proof | If b_W lies in U, then W⊆U, a contradiction. | 3.0 min | Low / Medium-high | C | A |
| AU100 | 2023 | Q3(e)(i) | 2 | L06,L03,L04 | Construct a map with nullity 2 whose image of the intersection does not have dimension 2. / Construction | Note the codomain error in the official solution [the source's correction is truncated]. | 3.0 min | Low / Medium-high | B | A |
| AU101 | 2023 | Q3(e)(ii) | 2 | L04,L06 | Prove no surjection can satisfy g(U)∩g(W)=0. / Proof | A surjection between equal-dimensional spaces is injective, while U∩W contains a nonzero vector. | 3.0 min | Low / High | C | A |
| AU102 | 2023 | Q3(e)(iii) | 2 | L06,L03 | Construct a rank-2 map with h(U)∩h(W)=0. / Construction | Map complementary directions to different basis vectors of the intersection. | 3.0 min | Low / High | B | A |
| AU103 | 2023 | Q4(a) | 8 | S04,M06 | Check a basis and apply Gram-Schmidt. / Definition/structure | Dot products, determinant and orthogonalisation. | 12.0 min | High / Medium/low | B | B |
| AU104 | 2023 | Q4(b) | 5 | G05,G09,P04 | Determine whether inner conjugation, left translation and the trivial map are homomorphisms, injective or surjective. / True-or-false/counterexample | Construct an inverse or counterexample. | 7.5 min | Medium / Medium/low | B | B |
| AU105 | 2023 | Q4(c) | 3 | G06,V09 | Prove each fibre of a surjective homomorphism is a left coset of kerφ. / Proof | Start with phi(g’)=phi(g) if and only if [source condition truncated]. | 4.5 min | Low / Medium-high | B | B |
| AU106 | 2023 | Q4(d) | 4 | G13,P07 | Prove z↦z^q is an automorphism of the group of p-th roots of unity. / Proof | A trivial kernel gives injectivity and finiteness gives surjectivity [following official-solution note truncated]. | 6.0 min | Medium-high / High | B | A |
| AU107 | 2023 | Q5(a)(i)-(iv) | 8 | G01,P01 | Determine whether four given sets are subgroups. / True-or-false/counterexample | Do not incorrectly separate powers in nonabelian groups. | 12.0 min | High / Medium/low | B | C |
| AU108 | 2023 | Q5(b) | 5 | P01 | Disjoint cycles, 2-cycles and sign. / Definition/structure | Trace cycles and count transposition parity. | 7.5 min | Medium / Medium/low | B | B |
| AU109 | 2023 | Q5(c) | 3 | G07 | List all cosets of a given order-2 subgroup of S3. / Definition/structure | Use index 3 and remove duplicates. | 4.5 min | Low / Medium-high | B | B |
| AU110 | 2023 | Q5(d) | 4 | P03,P04 | Prove dihedral-generator relations and change the generating set. / Proof | ara=r^{-1}, ab=r, and both inclusions. | 6.0 min | Medium-high / High | A | A |
| AU111 | 2023 | Q6(a) | 3 | G04,G01 | Prove a direct product of two groups is a group. / Proof | Identity and inverses coordinate by coordinate. | 4.5 min | Low / Medium/low | D | C |
| AU112 | 2023 | Q6(b) | 5 | G04 | Prove Z and Z^2 are not isomorphic. / Proof | Z is cyclic and Z^2 is not [source alternative argument truncated]. | 7.5 min | Medium-high / Medium-high | D | A |
| AU113 | 2023 | Q6(c)(i) | 2 | P05,P04 | Prove conjugation preserves commutators. / Proof | Insert g^{-1}g. | 3.0 min | Low / Medium/low | A | A |
| AU114 | 2023 | Q6(c)(ii) | 6 | P05,G08 | Prove products of commutators form a normal subgroup. / Proof | [a,b]^{-1}=[b,a]; conjugate each factor. | 9.0 min | Medium-high / Medium/low | A | A |
| AU115 | 2023 | Q6(c)(iii) | 4 | P05,G08 | Prove G/[G,G] is abelian. / Proof | (yx)^{-1}xy is a commutator. | 6.0 min | Medium-high / High | A | A |
| AU116 | 2024 | Q1(a) | 4 | M01 | Prove matrix distributivity from first principles. / Proof | Specify the (i,j) entry and summation indices. | 6.0 min | Medium-high / Medium/low | D | C |
| AU117 | 2024 | Q1(b)(i)-(iv) | 8 | M05,M12 | Decide four matrix propositions and give proofs/counterexamples. / Proof | Inverse-product formula, diagonal matrices need not be central, and counterexamples. | 12.0 min | High / Medium/low | A | A |
| AU118 | 2024 | Q1(c)(i)-(ii) | 4 | M13,M12 | Find the centres of two small matrix sets. / Calculation/solution | Scalar matrices or products identically zero. | 6.0 min | Medium / Medium-high | A | A |
| AU119 | 2024 | Q1(c)(iii) | 4 | M13,M12 | Find the centre of the strictly upper-triangular matrix algebra. / Calculation/solution | Test commutators with e_ij. | 6.0 min | Medium / High | A | A |
| AU120 | 2024 | Q2(a) | 3 | V02,V03 | Prove A∩B is a subspace. / Proof | Zero, addition and scalar multiplication. | 4.5 min | Low / Medium/low | B | B |
| AU121 | 2024 | Q2(b)(i) | 3 | V03,V06 | Deduce an intersection-dimension inequality from C⊆B. / Definition/structure | A∩C≤A∩B. | 4.5 min | Low / Medium/low | B | B |
| AU122 | 2024 | Q2(b)(ii) | 4 | V03,V06 | Prove equality of dimensions if and only if A∩B⊆C. / Proof | Inclusion and equal dimensions imply equality in finite dimensions. | 6.0 min | Medium-high / Medium-high | B | B |
| AU123 | 2024 | Q2(c) | 7 | V07,V06,L06 | Characterise infinite-dimensional V using intersections of subspaces. / Definition/structure | Choose independent vectors outside the finite-dimensional combined space [source construction truncated]. | 10.5 min | High / High | B | B |
| AU124 | 2024 | Q2(d) | 3 | V07,V08 | Give a concrete A’ in R[x] and determine possible numbers. / True-or-false/counterexample | Add higher-degree directions such as x^3. | 4.5 min | Low / Medium/low | C | B |
| AU125 | 2024 | Q3(a)(i) | 4 | L03,V04 | Find a basis for the kernel of a linear functional. / Calculation/solution | Parametrisation, independence and spanning. | 6.0 min | Medium / Medium/low | B | B |
| AU126 | 2024 | Q3(a)(ii) | 3 | L05 | Construct an isomorphism from R2 to ker f. / Construction | Give an explicit formula and inverse. | 4.5 min | Low / Medium/low | D | B |
| AU127 | 2024 | Q3(a)(iii) | 1 | L04 | Find the image dimension. / Calculation/solution | Domain dimension 3 minus nullity 2. | 1.5 min | Low / Medium/low | B | B |
| AU128 | 2024 | Q3(b)(i) | 3 | V10 | Deduce skew-symmetry from alternating bilinearity. / Definition/structure | rho(v+w,v+w)=0. | 4.5 min | Low / Medium-high | A | A |
| AU129 | 2024 | Q3(b)(ii) | 4 | V10 | Show the dot product is bilinear but not alternating. / True-or-false/counterexample | e1·e1=1. | 6.0 min | Medium / Medium/low | A | A |
| AU130 | 2024 | Q3(b)(iii) | 5 | V10,M06 | Construct an SL2 matrix for which v^TAw is alternating. / Construction | Take [[0,1],[-1,0]] and verify det=1. | 7.5 min | Medium-high / High | A | A |
| AU131 | 2024 | Q4(a) | 8 | L02,S01,S02 | Write the matrix, find eigenspaces and determine diagonalisability. / True-or-false/counterexample | The repeated root 1 has a two-dimensional eigenspace. | 12.0 min | High / Medium/low | C | B |
| AU132 | 2024 | Q4(b)(i) | 5 | M06,S09 | Derive a second-order recurrence for a tridiagonal determinant. / Definition/structure | Expand along the last row, then the last column. | 7.5 min | Medium / Medium/low | B | A |
| AU133 | 2024 | Q4(b)(ii) | 7 | S09,S01,S02 | Write the recurrence using a 2-by-2 matrix and diagonalise for a closed form. / Calculation/solution | Eigenvalues b,c and corresponding eigenvectors. | 10.5 min | High / High | B | A |
| AU134 | 2024 | Q5(a) | 2 | P01 | Define a cycle. / Definition/structure | Points outside its support are fixed. | 3.0 min | Low / Medium/low | C | C |
| AU135 | 2024 | Q5(b)(i) | 3 | P03,P01 | List the eight elements of D8 and their cycle decompositions. / Definition/structure | Powers of a rotation and reflections. | 4.5 min | Low / Medium/low | B | A |
| AU136 | 2024 | Q5(b)(ii) | 3 | P03,P04 | Prove that conjugating a rotation by a reflection gives its inverse. / Proof | στ=τσ^{-1}, then extend to powers. | 4.5 min | Low / Medium/low | B | B |
| AU137 | 2024 | Q5(b)(iii) | 5 | P03,G01 | Prove D is a subgroup without invoking known dihedral results. / Proof | Apply the subgroup test using gh^{-1}. | 7.5 min | Medium-high / Medium/low | A | A |
| AU138 | 2024 | Q5(c) | 4 | V09,G06,L03 | Prove the solution set of Ax=b is empty or a coset of kerA. / Proof | Take a particular solution u and write U=u+W. | 6.0 min | Medium-high / High | B | B |
| AU139 | 2024 | Q5(d) | 3 | G11,G07 | Subgroups of coprime order have trivial intersection. / Definition/structure | The intersection order divides both subgroup orders. | 4.5 min | Low / Medium-high | C | C |
| AU140 | 2024 | Q6(a) | 5 | G02 | Determine whether the additive groups Z, Q and M2(R) are cyclic. / True-or-false/counterexample | The source flags an incorrect official argument for M2's additive group, but the explanation is truncated. | 7.5 min | Medium / Medium/low | B | B |
| AU141 | 2024 | Q6(b) | 3 | G02,G05 | Prove a homomorphic image of a cyclic group is cyclic. / Proof | Imφ=<φ(g)>. | 4.5 min | Low / Medium/low | B | A |
| AU142 | 2024 | Q6(c) | 5 | P07,G05,G07 | Prove a nontrivial homomorphism from a prime-order group is injective. / Definition/structure | The kernel order is either 1 or p. | 7.5 min | Medium / Medium/low | A | A |
| AU143 | 2024 | Q6(d)(i)-(iii) | 3 | G09,G01 | Prove Aut(G) is a group under composition. / Proof | Verify the axioms individually. | 4.5 min | Low / Medium-high | A | A |
| AU144 | 2024 | Q6(d)(iv) | 4 | G09,P04,G05 | Construct the conjugation homomorphism C:G→Aut(G) and prove kerC=Z(G). / Construction | g↦(x↦gxg^{-1}). | 6.0 min | Medium-high / High | A | A |
| AU145 | 2025 | Q1(a) | 2 | M02 | Determine when a parameter matrix is in REF. / True-or-false/counterexample | Pivots move right and zero rows are placed correctly. | 3.0 min | Low / Medium/low | A | A |
| AU146 | 2025 | Q1(b) | 2 | M02 | Determine when it is in RREF. / True-or-false/counterexample | Pivots equal 1 and other entries in pivot columns vanish. | 3.0 min | Low / Medium/low | A | A |
| AU147 | 2025 | Q1(c) | 6 | M07,V04 | Find row-space and column-space bases in rank-1/rank-2 cases. / Calculation/solution | Determine whether the second row is zero or a multiple. | 9.0 min | Medium / Medium-high | A | A |
| AU148 | 2025 | Q1(d)(i) | 2 | M09,V03 | Prove row(A+B) is contained in row(A)+row(B). / Proof | Each row is the sum of the corresponding two rows. | 3.0 min | Low / Medium/low | A | A |
| AU149 | 2025 | Q1(d)(ii) | 2 | M09,L03 | Prove null(B) is contained in null(AB). / Proof | Bv=0⇒ABv=0. | 3.0 min | Low / Medium/low | B | B |
| AU150 | 2025 | Q1(d)(iii) | 3 | M09,M01 | Prove row(AB) is contained in row(B). / Proof | Every row of AB is a linear combination of rows of B. | 4.5 min | Low / Medium/low | A | A |
| AU151 | 2025 | Q1(d)(iv) | 3 | M09 | Disprove that AB=0 implies BA=0. / True-or-false/counterexample | Construct a minimal 2-by-2 example. | 4.5 min | Low / Medium-high | C | A |
| AU152 | 2025 | Q2(a) | 3 | V03,V02 | Show by example that the union of two subspaces need not be a subspace. / Definition/structure | The union of coordinate axes is not closed under addition. | 4.5 min | Low / Medium/low | C | A |
| AU153 | 2025 | Q2(b) | 5 | V03,V07 | Prove the union of an increasing subspace chain is a subspace. / Proof | Take max(n1,n2). | 7.5 min | Medium-high / Medium/low | A | A |
| AU154 | 2025 | Q2(c) | 1 | V07,V04 | Define finite dimension. / Definition/structure | A finite basis or finite generating set exists. | 1.5 min | Low / Medium/low | C | A |
| AU155 | 2025 | Q2(d) | 6 | V07,V06 | Prove an increasing subspace chain in finite dimensions eventually stabilises. / Proof | The finitely generated union space lies in some U_N. | 9.0 min | Medium-high / High | A | A |
| AU156 | 2025 | Q2(e) | 5 | V07,V08 | Construct a strictly increasing chain in F[x]. / Construction | Use polynomials of degree ≤n. | 7.5 min | Medium-high / Medium/low | B | A |
| AU157 | 2025 | Q3(a) | 3 | V08,L05 | Prove a function space on a finite set is in bijection with F4. / Proof | Four values uniquely determine the function. | 4.5 min | Low / Medium/low | B | A |
| AU158 | 2025 | Q3(b) | 6 | V01,V08,L05 | Transfer a vector-space structure through a bijection and prove an isomorphism. / Proof | Prove preservation of addition and scalar multiplication. | 9.0 min | Medium-high / High | C | A |
| AU159 | 2025 | Q3(c) | 3 | V04,L05 | Pull back the standard basis to obtain a function-space basis. / Definition/structure | Use indicator functions. | 4.5 min | Low / High | C | A |
| AU160 | 2025 | Q3(d) | 4 | L02,L01 | Find the matrix of the rotation-induced composition operator. / Calculation/solution | f_{a,b}∘R=f_{R^{-1}(a,b)}. | 6.0 min | Medium / Medium/low | C | A |
| AU161 | 2025 | Q3(e) | 4 | L03,L04,L05 | Find the kernel, image and rank of the invertible induced operator. / Calculation/solution | The inverse corresponds to rotation in the opposite direction. | 6.0 min | Medium / Medium/low | B | A |
| AU162 | 2025 | Q4(a) | 11 | L02,S01,S02 | Write the matrix, find three eigenspaces and diagonalise. / Calculation/solution | There are three distinct real eigenvalues. | 16.5 min | High / Medium/low | C | A |
| AU163 | 2025 | Q4(b)(i) | 5 | M08,M06 | Prove rank A<k if and only if all k-order minors vanish. / Proof | Choose independent rows and columns to form a nonzero minor. | 7.5 min | Medium-high / Medium-high | A | A |
| AU164 | 2025 | Q4(b)(ii) | 4 | M08,V06 | Fix a nonzero k-order minor and characterise using containing (k+1)-order minors [source conclusion truncated]. / Definition/structure | Extend the independent rows, then extend the columns. | 6.0 min | Medium / High | A | B |
| AU165 | 2025 | Q5(a)(i) | 4 | S04 | Apply Gram-Schmidt to matrix columns and normalise. / Definition/structure | Use orthogonal projection. | 6.0 min | Medium / Medium/low | A | A |
| AU166 | 2025 | Q5(a)(ii)-(iii) | 6 | S06,S04 | Express original columns in orthonormal coordinates and form QR. / Calculation/solution | Q has orthonormal columns and R is upper triangular. | 9.0 min | Medium / Medium/low | A | A |
| AU167 | 2025 | Q5(b)(i) | 3 | S12,G01 | Prove integer orthogonal matrices form a subgroup. / Proof | The inverse is the transpose and integrality is preserved. | 4.5 min | Low / Medium/low | A | A |
| AU168 | 2025 | Q5(b)(ii) | 3 | S12 | Prove permutation matrices belong to O_n(Z). / Proof | They permute the standard basis. | 4.5 min | Low / Medium-high | A | A |
| AU169 | 2025 | Q5(b)(iii) | 4 | S12,G04 | Characterise O_n(Z) as signed permutation matrices and find its order 2^n n!. / Calculation/solution | Each column has one ±1, in distinct rows. | 6.0 min | Medium / High | A | A |
| AU170 | 2025 | Q6(a)(i) | 2 | P01,P02 | Simplify a permutation product and find its order. / Calculation/solution | Disjoint cycles and lcm. | 3.0 min | Low / Medium/low | B | A |
| AU171 | 2025 | Q6(a)(ii) | 4 | P02 | Find the maximum element order in S9. / Calculation/solution | Compare the lcm values of the partitions of 9. | 6.0 min | Medium / Medium/low | B | A |
| AU172 | 2025 | Q6(b)(i)-(iii) | 7 | G02,G04 | Determine whether S5, the shear-matrix group and regular-polygon rotation group are cyclic. / True-or-false/counterexample | Nonabelian structure, isomorphism with Z, or a basic rotation, respectively. | 10.5 min | High / Medium/low | B | A |
| AU173 | 2025 | Q6(c) | 7 | G12,G01 | Prove a nonempty multiplicatively closed set is a subgroup when every element has finite order. / Proof | h^n=e and h^{-1}=h^{n-1}. | 10.5 min | High / Medium-high | A | A |
| AU174 | 2026 | Q1(a) | 3 | L02 | Write the standard-basis matrix of D. / Calculation/solution | Read columns from the coordinate formula. | 4.5 min | Low / Medium/low | A | A |
| AU175 | 2026 | Q1(b) | 6 | L03,V04 | Find a parametrisation, basis and dimension for kerD. / Calculation/solution | Solve the equations and extract two parameter directions. | 9.0 min | Medium / Medium/low | A | A |
| AU176 | 2026 | Q1(c) | 3 | L02,L03,L04 | Write the matrix of G; find kerG and dim imG. / Calculation/solution | s=t=u=r. | 4.5 min | Low / Medium/low | A | A |
| AU177 | 2026 | Q1(d) | 4 | V03,L03,L12 | Prove kerD∩imG=0 when the characteristic is not 2. / Proof | Invertibility of 2 eliminates the parameters. | 6.0 min | Medium-high / Medium-high | A | A |
| AU178 | 2026 | Q1(e) | 2 | V03,V06,L04 | Use dimensions to deduce F5=kerD+imG and find rankD. / Calculation/solution | 2+3=5 and the intersection is zero. | 3.0 min | Low / Medium/low | A | A |
| AU179 | 2026 | Q1(f) | 2 | L12,V03 | Construct a nonzero intersection element in characteristic 2. / Construction | Use 1=-1. | 3.0 min | Low / High | A | A |
| AU180 | 2026 | Q2(a) | 3 | V08,L01,L02 | Prove J:p(x)↦p(-x) is linear and write its matrix. / Proof | Odd-degree terms change sign. | 4.5 min | Low / Medium/low | A | A |
| AU181 | 2026 | Q2(b) | 7 | V02,V04,L11,L12 | Find bases of the even/odd subspaces, their intersection and the characteristic-2 cases. / Calculation/solution | V+=ker(I-J), V-=ker(I+J). | 10.5 min | High / Medium/low | A | A |
| AU182 | 2026 | Q2(c) | 2 | L02,V08 | Write the differentiation-operator matrix. / Calculation/solution | D(1), D(x), D(x2), D(x3). | 3.0 min | Low / Medium/low | C | C |
| AU183 | 2026 | Q2(d) | 4 | L09,L10 | Prove DJ=-JD and deduce D exchanges the two subspaces. / Proof | Chain rule / formal derivative. | 6.0 min | Medium-high / Medium-high | A | C |
| AU184 | 2026 | Q2(e) | 4 | L10,L12,L04 | Find restricted-map ranks in characteristic 2/3. / Calculation/solution | Find images of basis vectors in each subspace. | 6.0 min | Medium / High | A | A |
| AU185 | 2026 | Q3(a)(i) | 3 | S01,S03 | Find the eigenvalues of a symmetric matrix. / Calculation/solution | Use its symmetric structure. | 4.5 min | Low / Medium/low | A | A |
| AU186 | 2026 | Q3(a)(ii) | 4 | S03,S04 | Construct orthogonal Q and diagonal D. / Construction | Orthogonalise within a repeated eigenspace. | 6.0 min | Medium-high / Medium/low | A | A |
| AU187 | 2026 | Q3(a)(iii) | 3 | S07,S03 | Construct B with A=BB^T. / Construction | B=Q sqrt(D). | 4.5 min | Low / Medium/low | A | A |
| AU188 | 2026 | Q3(b)(i) | 3 | S11,M12,S05 | Deduce orthogonality of the real block matrix from unitary Z=A+iB. / Proof | Z*Z=I. | 4.5 min | Low / Medium/low | A | A |
| AU189 | 2026 | Q3(b)(ii) | 3 | M12 | Prove the block matrix commutes with J. / Proof | JM=MJ. | 4.5 min | Low / Medium-high | B | B |
| AU190 | 2026 | Q3(b)(iii) | 4 | M12,S11,S05 | Prove an orthogonal matrix commuting with J arises from a unitary matrix. / Proof | General blocks → required block form → unitarity condition. | 6.0 min | Medium-high / High | A | A |
| AU191 | 2026 | Q4(a)(i)-(iii) | 8 | P01 | Disjoint cycles, transposition decomposition and sign. / Definition/structure | Trace images, split cycles and count transpositions. | 12.0 min | High / Medium/low | A | A |
| AU192 | 2026 | Q4(b)(i) | 3 | P04 | Prove conjugacy in a group is an equivalence relation. / Proof | Identity, inverse conjugation and composition of conjugations. | 4.5 min | Low / Medium/low | A | A |
| AU193 | 2026 | Q4(b)(ii) | 2 | P04 | Prove conjugation is compatible with positive integer powers. / Proof | Insert g^{-1}g. | 3.0 min | Low / Medium/low | A | A |
| AU194 | 2026 | Q4(c) | 7 | P04,P02 | Prove two permutations in S_n are conjugate if and only if cycle types agree. / Proof | Conjugation relabels cycles; conversely construct the relabelling. Source annotations: C/D, unseen, 2023 Q5(d). | 10.5 min | High / Medium-high | A | A |
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