21 MATH40003 Practice Paper 1 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: 294b1845d2b89f2111aa57c6173dcaf4fb31177f80001b4e1b1a8a29fdfda062
Source date: 2026-08-06

Source page 1

Source page 2

Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

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.

Source page 3

Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

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 ).

Source page 4

Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

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.

Source page 5

Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

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.