23 MATH40003 Practice Paper 3 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: 4399948a19e3ffde47a1a4b3cf8fdd184eef6f8937d877424ad8128008079f21
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: 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.

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: 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 .

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: 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.

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: 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.