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MATH40003 Seven Year Effective Coverage Matrix REVISED

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: c938ab40f5a416e08810d0edcd87c91aa4a13279eb67dde8356345153c379a8c
Source date: 2026-08-06

EffectiveCoverage

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unit_idyearquestionpartmarkscodesdescriptionN1Q1N1Q2N1Q3N1Q4N2Q1N2Q2N2Q3N2Q4N3Q1N3Q2N3Q3N3Q4N4Q1N4Q2N4Q3N4Q4N5Q1N5Q2N5Q3N5Q4N6Q1N6Q2N6Q3N6Q4BEST
AU00120201(a)2M02,M05use elementary row operations to find the given 3 the inverse of a matrix of this orderDDDDBDDDDDDDDDDDDDDDDDDDB
AU00220201(b)-(c)2S05,M05define an orthogonal matrix and verify A^{-1}=A^TDDDDDDDDDDDDDDDDDDADDDDDA
AU00320201(d)4L02,S05derive the two-dimensional rotation matrix from images of basis vectorsBCDDDDBDDDDDDDDDDDADCDDDA
AU00420201(e)3S05write two reflection matrices and prove their composition is a rotationDDDDDDDDDDDDDDDDDDADDDDDA
AU00520201(f)7S05,M06classify all real 2 orthogonal matrices of this order are rotations or reflectionsDDDDDDDDDDCDDDDDCCADDDDDA
AU00620201(g)2L02,S05matrix of a three-dimensional rotation about a given axis in an orthonormal basisBCDDDDCDDDDDDDDDDDADCDDDA
AU00720202(a)1M04write the augmented matrix of the linear systemADDDDDDDDDDDCDDDDDDDDDDDA
AU00820202(b)6M03,M04by parameter case a,b classify existence of solutions and find all solutionsADDDDDDDDDDDCDDDDDDDDDDDA
AU00920202(c)5M07,M04define the row space / row rank, and find A and (A|c) the row space and rank ofCDDDBDDCDDDDCDDDDDDDDDDDB
AU01020202(d)8V09,L03,L04,M04prove the solution set of a consistent system has dimension n-rank(A) a coset of the subspace ofBDDDDDBDDCDDADDDDDDDCDDDA
AU01120203(a)3V02state the subspace criterionDDDDDADDDCDDDDDDDDDDDDDDA
AU01220203(b)4V02,V04,V08prove that polynomials with coefficients satisfying the recurrence form a subspace and find a basisDCDDCBCCDCDDDDDDCDDDCDDDB
AU01320203(c)8L01,L03,V02define linear maps, kernels and images, and prove the kernel is a subspaceBCDDDCBDDCDDCCDDDDDDCDDDB
AU01420203(d)5L01,L03,L10,V03verify T_n linear; find image and kernel and determine U_n whether it isBCDDDCBDDBDDCCDDDDDDCCDDB
AU01520204(a)(i)3M11,M05From A^2-3A+2I=0 prove A invertible, and write its inverseDDDDDDDDDDDDDDDDDDDDDADDA
AU01620204(a)(ii)-(iii)5S01,S02,V03prove E1 and E2 the direct sum is the whole space, implying diagonalisabilityBDCDDCDBDCDDDBCDCDDDDACDA
AU01720204(a)(iv)2S01,L10From A^k v boundedness implies v belongs to E1DBCDDDCCDDDDDDCDCDDDCCCDB
AU01820204(a)(v)2S01,S02,M10list 2 three similarity normal forms in this dimensionDCCDDDDBDDDDDDCDCDDDDACDA
AU01920204(b)(i)2S07prove B^TB positive definiteDDCDDDDDDDDDDDDDDDDDDDADA
AU02020204(b)(ii)6S03,S08,S07derive from the spectral theorem the invertible matrix's SVD type decomposition B=QDPDDADDDDDDDDDDDBDDDDDDDADA
AU02120205(a)6M06define the determinant recursively and prove scaling one row scales the determinant by the same factorDDDDDDDDDDCDDDDDACDDDDDDA
AU02220205(b)3M06use multiplicativity and transpose properties to calculate a composite matrix determinantDDDDDDDDDDCDDDDDACDDDDDDA
AU02320205(c)(i)-(iii)7G07define left cosets; prove cosets of the same subgroup are equal or disjoint andDDDDDDDDDDDDDDDBDDDDDDDDB
AU02420205(c)(iv)4G07,G11characterise intersections of cosets of different subgroups and give two types of exampleDDDDDDDDDDDDDDDBDDDDDDDDB
AU02520206(a)(i)-(iii)6G02,G03define element order; analyse 12 numbers of elements of each order in a cyclic group of this orderDDDDDDDDDDDDDDDADDDCDDDCA
AU02620206(a)(iv)3G07,G11find a proper subgroup A,B make the intersection trivial and AB=GDDDDDDDDDDDDDDDADDDDDDDDA
AU02720206(b)6G05,G06,G01define homomorphism and kernel; prove the kernel is a subgroup and equal images occur precisely in the same cosetDDDDDDDDCDDCBDDADDCCDDDDA
AU02820206(c)5G05,G13construct four types of nontrivial homomorphismDDDDDDDDDDDDDDDADDDCDDDDA
AU02920211(a)10M03,M04by parameter case alpha classify linear systems and give the unique or infinitely many solutionsADDDDDDDDDDDCDDDDDDDDDDDA
AU03020211(b)2V02use only vector-space axioms to prove a set without zero is not a subspaceDDDDDCDDDCDDDDDDDDDDDDDDC
AU03120211(c)(i)2V02,V09determine the original nonhomogeneous solution set S whether it is a subspaceDDDDDCDDDCDDADDDDDDDDDDDA
AU03220211(c)(ii)3V03,V09analyse the solution set after a specified translation and determine whether it is a subspaceCDDDDCDDDCDDACDDDDDDDDDDA
AU03320211(c)(iii)3V03,V09analyse the solution set after another translation and determine whether it is a subspaceCDDDDCDDDCDDACDDDDDDDDDDA
AU03420212(a)3V02,V08prove the parametrised set W is R4 subspaceDADDDCCDDCDDDDDDDDDDCDDDA
AU03520212(b)5V04,V05prove the four given vectors form a basis and find a general vector's B coordinatesDBDDCDCCDCDDDDDDCDDDDDDDB
AU03620212(c)5L02,L03,L04,V05find the image, image dimension and nonstandard-basis matrix of a linear mapBCDDDDBDDCDDCDDDDDDDCDDDB
AU03720212(d)(i)4L01prove the composite operator T*:f↦T∘f linearDCDDDDBDDDDDDCDDDDDDBDDDB
AU03820212(d)(ii)3L04,L06find T* the rank of the matrixBDDDDDBDDBDDCDDDDDDDDDDDB
AU03920213(a)11L01,L07determine whether five maps are linear and idempotentDADDDDBDDDDDDADDDDDDCCDDA
AU04020213(b)2L01prove a composite of linear maps is linearDADDDDCDDDDDDCDDDDDDCDDDA
AU04120213(c)4L07,L09determine whether a composite of two idempotents must be idempotent and give a counterexampleDADDDDDDDDDDDADDDDDDDCDDA
AU04220213(d)3L07,V03discuss idempotence T satisfies ImT=kerT the possibility ofBADDDCDDDCDDDADDDDDDDCDDA
AU04320214(a)(i)6S01,S09write a two-dimensional recurrence as a matrix and find eigenvalues / vectorDDCDDDDCDDDDDDADCCDDDCCDA
AU04420214(a)(ii)-(iii)4S09,S02find the general term and recover initial values from later valuesDDCDDDDBDDDDDDADDBDDDCCDA
AU04520214(b)(i)2S04prove the squared-norm formula for a linear combination of orthogonal vectorsDDCDDDDDDDDDDDCDDDCDDDADA
AU04620214(b)(ii)8S03,S10,S04find, for a real symmetric matrix on the unit sphere, ||Av|| maximum and minimumDDBDDDDDDDDDDDADDDCDDDADA
AU04720215(a)(i)3M06prove the determinant of a matrix with exactly one nonzero entry in every row and column is ± nonzeroDDDDDDDDDDCDDDDDACDDDDDDA
AU04820215(a)(ii)3M06replace a zero entry by 1 then explain why the determinant is unchangedDDDDDDDDDDCDDDDDACDDDDDDA
AU04920215(b)6S01,M08from eigenspace dimensions 2 prove the characteristic polynomial containsDDCDCDDBDDDDDDCDADDDDBCDA
AU05020215(c)(i)3G01,G07prove the stabilising set H is S6 subgroup, and find its orderDDDDDDDDCDDCDDDCDDCBDDDDB
AU05120215(c)(ii)-(iii)5G07use g(1),g(2) use unordered images to characterise cosets and find the index in two waysDDDDDDDDDDDDDDDBDDDDDDDDB
AU05220216(a)8P06,G01determine whether the set of squares is a subgroup in four group settings / equalsDDDDDDDDCDDCDDDDDDCADDDDA
AU05320216(b)(i)4P01,P02calculation S9 disjoint cycles and orders of several composed permutations inDDDBDDDDDDDDDDDDDDDCCDDAA
AU05420216(b)(ii)4P02list S9 in 3 cycle types of elements of this order and their countDDDBDDDDDDDDDDDDDDDADDDCA
AU05520216(c)4G03,P07prove that in a finite group p the number of elements of this order is p-1 dividesDDDDDDDDDDDDDDDCDDDADDDDA
AU05620221(a)(i)-(iii)8M03,L12the same linear system over each of R, F5, F7 solve overCCDDDDDDDDDDADDDDDDDCCDDA
AU05720221(b)4V01prove additive cancellation from vector-space axiomsDDDDDDCDDDDDDDDDDDDDDDDDC
AU05820221(c)(i)4V01verify V×W several vector-space axioms forDDDDDDCDDDDDDDDDDDDDDDDDC
AU05920221(c)(ii)4V01,V02prove A×B is V×W subspaceDDDDDCCDDCDDDDDDDDDDDDDDC
AU06020222(a)4V04,V08prove the four given 2 matrices of this order form M2(R) a basis ofDADDCCCCDCDDDDDDBDDDCDDDA
AU06120222(b)(i)-(ii)4L02,V05find the transpose operator's matrix in two basesCADDDDCDDDDDDDDDDDDDCDDDA
AU06220222(b)(iii)3V05,M10find the change-of-basis matrix of the identity mapDADDDDCDDDDDDDDDDDDDDDDDA
AU06320222(c)7L02,L01prove the matrix in some basis is I then the map itself is the identityCCDDDDCDDDDDDCDDDDDDCDDDC
AU06420222(d)2L03,L04find the rank and nullity of a linear functionalBDDDDDBDDBDDCDDDDDDDCDDDB
AU06520223(a)2L08,L02prove the shift operator is nilpotentCADDDDCDDDDDDADDDDDDCDDDA
AU06620223(b)4L08,V02prove Nil(T) is a subspaceDADDDCDDDCDDDCDDDDDDDDDDA
AU06720223(c)8L08,V06prove the nilpotency index is at most dimVCADDDBDDDCDDDADDDDDDDDDDA
AU06820223(d)6L09prove I-T and I+T invertibleDADDDDDDDDDDDADDDDDDDADDA
AU06920224(a)(i)2L02,V05write a matrix from images of basis vectors in a nonstandard basisBCDDDDBDDDDDDDDDDDDDCDDDB
AU07020224(a)(ii)6S01,S02find eigenspaces and show that insufficient geometric multiplicity prevents diagonalisationDDCDDDDBDDDDDDCDBDDDDBCDB
AU07120224(b)(i)3M06,M05find the parameter matrix's determinant and determine invertibilityDDDDDDDDDDCDDDDDBCDDDDDDB
AU07220224(b)(ii)2M10,M06rule out existence using equality of determinants of similar matrices PDCDDDDDDDDCDDDDDBCDDDDDDB
AU07320224(c)7S04,S06prove existence for every invertible real matrix QR decompositionDDCDDDDDDDDDDDCDDDADDDADA
AU07420225(a)(i)3S04check that three vectors form an orthonormal basisDDADDDDDDDDDDDCDDDCDDDCDA
AU07520225(a)(ii)3S03From T(u)·v=u·T(v) prove the matrix is symmetricDDADDDDDDDDDDDCDDDDDDDCDA
AU07620225(a)(iii)2S03,S01use the spectral theorem to prove the characteristic polynomial of a symmetric matrix over R splittingDDBDDDDCDDDDDDBDCDDDDCADA
AU07720225(b)3S05prove the two-dimensional rotation matrix is orthogonalDDDDDDDDDDDDDDDDDDADDDDDA
AU07820225(c)(i)2S12prove a permutation matrix is orthogonalDDDDDDDDDDDDDDDDDDADDDDDA
AU07920225(c)(ii)7S12,G01,G04,P01prove permutation matrices form a subgroup and with S_n isomorphismDDDADDDDCDDCDDDDDDCCCDDCA
AU08020226(a)(i)2G01prove R2 a line is an additive subgroup ofDDDDDDDDCDDCDDDDDDCCDDDDC
AU08120226(a)(iii)2G04construct this subgroup and (R,+) isomorphismDDDDDDDDDDDDDDDDDDDDDDDCC
AU08220226(a)(iv)4G08,G07define an operation on cosets and prove the quotient group is isomorphic to RDDDDDDDDDDDCDDDBDDDDDDDDB
AU08320226(b)3P01convert two-line notation into disjoint cyclesDDDBDDDDDDDDDDDDDDDDCDDCB
AU08420226(c)(i)-(ii)5G08,G07,G05prove the index 2 the subgroup is normal, and apply to A5=ker(sgn)DDDDDDDDDDDCDDDBDDDDDDDDB
AU08520226(d)4G10prove every finite group embeds in some S_nDDDDDDDDDDDDDDDDDDDADDDDA
AU08620231(a)3M05use repeated rows to show the matrix is not invertibleDDDDBDDDDDDDDDDDDDDDDDDDB
AU08720231(b)4M01prove matrix-multiplication distributivity from the definitionDDDDDDDDADDDDDDDDDDDDDDDA
AU08820231(c)6M13,M01prove M_n(R) forms a ring under the usual operationsDDDDDDDDADDDDDDDDDDDDDDDA
AU08920231(d)(i)-(ii)4M13show two custom operations do not form a ring and give counterexamplesDDDDDDDDADDDDDDDDDDDDDDDA
AU09020231(e)3M13construct a binary operation failing both left and right distributivityDDDDDDDDCDDDDDDDDDDDDDDDC
AU09120232(a)6M05,M03contains parameters 3 invertibility conditions and inverse for matrices of this orderCDDDDDDDDDDDCDDDDDDDDDDDC
AU09220232(b)6M03take eye colour / translate the teacher/student counts into 5 a variable system and solveCDDDDDDDDDDDCDDDDDDDDDDDC
AU09320232(c)(i)2G03find the finite-order elements in the additive group of a real vector spaceDDDDDDDDDDDDDDDCDDDCDDDDC
AU09420232(c)(ii)3G01,V02disprove that every additive subgroup is a subspaceDDDDDCDDCCDCDDDDDDCCDDDDC
AU09520232(d)3G01,V03prove the union of two subgroups, neither containing the other, is not a subgroupCDDDDCDDCCDCDCDDDDCCDDDDC
AU09620233(a)3V02,V03prove U∩W is a subspaceBDDDDCDDDADDDCDDDDDDDDDDA
AU09720233(b)3V04,V06prove the two given vectors are U∩W basisBDDDCCCCDADDDCDDCDDDDDDDA
AU09820233(c)6V06,V04extend a basis of the intersection to a basis of each U and W a basis ofBDDDCCCCDADDDCDDCDDDDDDDA
AU09920233(d)2V03,V06prove any choice of complementary vectors together forms V a basis ofADDDDCDDDCDDDCDDDDDDDDDDA
AU10020233(e)(i)2L06,L03,L04construct nullity 2 and the image dimension of the intersection is not 2 a map ofBDDDDDBDDADDCDDDDDDDCDDDA
AU10120233(e)(ii)2L04,L06prove there is no surjection such that g(U)∩g(W)=0CDDDDDCDDADDCDDDDDDDDDDDA
AU10220233(e)(iii)2L06,L03construct rank 2 and h(U)∩h(W)=0 a map ofBDDDDDBDDADDCDDDDDDDCDDDA
AU10320234(a)8S04,M06verify a basis and perform Gram-SchmidtDDCDDDDDDDCDDDCDBCBDDDBDB
AU10420234(b)5G05,G09,P04determine whether inner conjugation, left translation and the trivial map are homomorphisms, injective or surjectiveDDDBDDDDDDDBDDDBDDDDDDDDB
AU10520234(c)3G06,V09prove each fibre of a surjective homomorphism is kerφ left cosetDDDDDDDDDDDDBDDBDDDDDDDDB
AU10620234(d)4G13,P07prove z↦z^q is p automorphisms of the group of roots of unity of this orderDDDDDDDDDDDDDDDBDDDADDDDA
AU10720235(a)(i)-(iv)8G01,P01determine whether four given sets are subgroupsDDDCDDDDCDDCDDDDDDCCCDDCC
AU10820235(b)5P01disjoint cycles , 2- cycles and signDDDBDDDDDDDDDDDDDDDDCDDCB
AU10920235(c)3G07list S3 given in 2 all cosets of a subgroup of this orderDDDDDDDDDDDDDDDBDDDDDDDDB
AU11020235(d)4P03,P04prove dihedral-generator relations and change the generating setDDDBDDDDDDDADDDDDDDDDDDDA
AU11120236(a)3G04,G01prove the direct product of two groups is a groupDDDDDDDDCDDCDDDDDDCCDDDCC
AU11220236(b)5G04prove Z and Z^2 not isomorphicDDDDDDDDDDDDDDDDDDDDDDDAA
AU11320236(c)(i)2P05,P04prove conjugation preserves commutatorsDDDBDDDDDDDADDDDDDDDDDDDA
AU11420236(c)(ii)6P05,G08prove the set of products of commutators is a normal subgroupDDDDDDDDDDDADDDDDDDDDDDDA
AU11520236(c)(iii)4P05,G08prove G/[G,G] commutativityDDDDDDDDDDDADDDDDDDDDDDDA
AU11620241(a)4M01prove matrix distributivity from first principlesDDDDDDDDCDDDDDDDDDDDDDDDC
AU11720241(b)(i)-(iv)8M05,M12truth values and proofs of four matrix propositions / counterexampleDDADDDDDCDCDDDDDDDDDDDDDA
AU11820241(c)(i)-(ii)4M13,M12find the centres of two small sets of matricesDDCDDDDDADCDDDDDDDDDDDDDA
AU11920241(c)(iii)4M13,M12find the centre of the strictly upper-triangular matrix algebraDDCDDDDDADCDDDDDDDDDDDDDA
AU12020242(a)3V02,V03prove A∩B is a subspaceBDDDDCDDDBDDDCDDDDDDDDDDB
AU12120242(b)(i)3V03,V06From C⊆B deduce a dimension inequality for intersectionsBDDDDCDDDBDDDCDDDDDDDDDDB
AU12220242(b)(ii)4V03,V06prove equal dimensions if and only if A∩B⊆CBDDDDBDDDBDDDCDDDDDDDDDDB
AU12320242(c)7V07,V06,L06characterise using intersections of subspaces V infinite-dimensionalBDDDDBDDDCDDDCDDDDDDDDDDB
AU12420242(d)3V07,V08in R[x] give a concrete example in A’ and determine possible numbersDCDDDBCDDDDDDDDDDDDDCDDDB
AU12520243(a)(i)4L03,V04find a basis for the kernel of a linear functionalBDDDCDBCDCDDCDDDCDDDCDDDB
AU12620243(a)(ii)3L05construct R2 to ker f an isomorphism ofDDDDDDBDDDDDDDDDDDDDDDDDB
AU12720243(a)(iii)1L04find the image dimensionBDDDDDBDDBDDCDDDDDDDDDDDB
AU12820243(b)(i)3V10deduce skew-symmetry from alternating bilinearityDDDDDDDDDDADDDDDDDDDDDDDA
AU12920243(b)(ii)4V10show the dot product is bilinear but not alternatingDDDDDDDDDDADDDDDDDDDDDDDA
AU13020243(b)(iii)5V10,M06construct SL2 a matrix such that v^TAw alternatingDDDDDDDDDDADDDDDCCDDDDDDA
AU13120244(a)8L02,S01,S02write the matrix, find eigenspaces and determine diagonalisabilityCCCDDDCBDDDDDDCDCDDDCBCDB
AU13220244(b)(i)5M06,S09derive a second-order recurrence for a tridiagonal determinantDDDDDDDDDDCDDDBDBADDDDDDA
AU13320244(b)(ii)7S09,S01,S02write the recurrence as 2 matrix of this order and diagonalise to obtain a closed formDDCDDDDBDDDDDDBDCADDDCCDA
AU13420245(a)2P01define a cyclicDDDCDDDDDDDDDDDDDDDDCDDCC
AU13520245(b)(i)3P03,P01list D8 the eight elements and their cycle decompositionsDDDCDDDDDDDADDDDDDDDCDDCA
AU13620245(b)(ii)3P03,P04prove conjugation of a rotation by a reflection yields its inverseDDDBDDDDDDDCDDDDDDDDDDDDB
AU13720245(b)(iii)5P03,G01prove without using known dihedral results D is a subgroupDDDDDDDDCDDADDDDDDCCDDDDA
AU13820245(c)4V09,G06,L03prove Ax=b the solution set is empty or is kerA the coset ofBDDDDDCDDCDDBDDBDDDDCDDDB
AU13920245(d)3G11,G07subgroups of coprime order have trivial intersectionDDDDDDDDDDDDDDDCDDDDDDDDC
AU14020246(a)5G02determine Z, Q, M2(R) whether the additive group is cyclicDDDDDDDDDDDDDDDCDDDDDDDBB
AU14120246(b)3G02,G05prove a homomorphic image of a cyclic group is cyclicDDDDDDDDDDDDDDDADDDDDDDCA
AU14220246(c)5P07,G05,G07a nontrivial homomorphism from a prime-order group is injectiveDDDDDDDDDDDDDDDADDDCDDDDA
AU14320246(d)(i)-(iii)3G09,G01prove Aut(G) forms a group under compositionDDDCDDDDCDDADDDDDDCCDDDDA
AU14420246(d)(iv)4G09,P04,G05construct the conjugation homomorphism C:G→Aut(G) and kerC=Z(G)DDDBDDDDDDDADDDADDDDDDDDA
AU14520251(a)2M02determine when the parameter matrix REFDDDDADDDDDDDDDDDDDDDDDDDA
AU14620251(b)2M02determine when RREFDDDDADDDDDDDDDDDDDDDDDDDA
AU14720251(c)6M07,V04by rank 1/2 find bases for row and column spaces by casesCDDDADCCDBDDDDDDCDDDDDDDA
AU14820251(d)(i)2M09,V03prove row(A+B) contained in row(A)+row(B)CDDDACDDDCDDDCDDDDDDDDDDA
AU14920251(d)(ii)2M09,L03prove null(B) contained in null(AB)BDDDCDBDDCDDCDDDDDDDCDDDB
AU15020251(d)(iii)3M09,M01prove row(AB) contained in row(B)DDDDADDDCDDDDDDDDDDDDDDDA
AU15120251(d)(iv)3M09disprove AB=0 deduce BA=0DDDDADDDDDDDDDDDDDDDDDDDA
AU15220252(a)3V03,V02give an example showing the union of two subspaces need not be a subspaceCDDDDADDDCDDDCDDDDDDDDDDA
AU15320252(b)5V03,V07prove the union of an increasing subspace chain is a subspaceBDDDDADDDCDDDCDDDDDDDDDDA
AU15420252(c)1V07,V04define finite dimensionDDDDCACCDCDDDDDDCDDDDDDDA
AU15520252(d)6V07,V06prove that an increasing chain of subspaces of a finite-dimensional space eventually stabilisesCDDDDADDDCDDDCDDDDDDDDDDA
AU15620252(e)5V07,V08in F[x] construct a strictly increasing chainDCDDDACDDDDDDDDDDDDDCDDDA
AU15720253(a)3V08,L05prove the function space on a finite set and F4 bijectionDCDDDCADDDDDDDDDDDDDADDDA
AU15820253(b)6V01,V08,L05transfer the vector-space structure through a bijection and prove an isomorphismDCDDDCADDDDDDDDDDDDDCDDDA
AU15920253(c)3V04,L05pull the standard basis back to a basis of the function spaceDDDDCDACDCDDDDDDCDDDDDDDA
AU16020253(d)4L02,L01find the matrix of the rotation-induced function-composition operatorCCDDDDADDDDDDCDDDDDDCDDDA
AU16120253(e)4L03,L04,L05find the kernel, image and rank of the invertible induced operatorBDDDDDADDBDDCDDDDDDDCDDDA
AU16220254(a)11L02,S01,S02write the matrix, find three eigenspaces and diagonaliseCCCDDDCADDDDDDCDCDDDCBCDA
AU16320254(b)(i)5M08,M06prove rank A<k if and only if all k minors of this order are 0DDDDBDDADDCDDDDDACDDDDDDA
AU16420254(b)(ii)4M08,V06fix a nonzero k minors of this order, using those containing it k+1 characterisation by minors of this orderCDDDCCDCDBDDDCDDCDDDDDDDB
AU16520255(a)(i)4S04apply to the matrix columns Gram-Schmidt and normaliseDDCDDDDDDDDDDDCDDDADDDCDA
AU16620255(a)(ii)-(iii)6S06,S04write the original columns in orthonormal-basis coordinates and form QRDDCDDDDDDDDDDDCDDDADDDADA
AU16720255(b)(i)3S12,G01prove integer orthogonal matrices form a subgroupDDDDDDDDCDDCDDDDDDACDDDDA
AU16820255(b)(ii)3S12prove the permutation matrix belongs to O_n(Z)DDDDDDDDDDDDDDDDDDADDDDDA
AU16920255(b)(iii)4S12,G04characterise O_n(Z) is a signed permutation matrix, and find the order 2^n n!DDDDDDDDDDDDDDDDDDADDDDBA
AU17020256(a)(i)2P01,P02simplify the permutation product and find its orderDDDBDDDDDDDDDDDDDDDCCDDAA
AU17120256(a)(ii)4P02find S9 maximum element order inDDDBDDDDDDDDDDDDDDDCDDDAA
AU17220256(b)(i)-(iii)7G02,G04determine S5, whether the shear-matrix group and regular-polygon rotation group are cyclicDDDDDDDDDDDDDDDBDDDDDDDAA
AU17320256(c)7G12,G01prove that when every element has finite order, a nonempty multiplicatively closed set is automatically a subgroupDDDDDDDDCDDCDDDDDDCADDDAA
AU17420261(a)3L02write D the standard-basis matrix ofACDDDDCDDDDDDDDDDDDDCDDDA
AU17520261(b)6L03,V04find kerD parametric form, basis and dimension ofADDDCDBCDCDDCDDDCDDDCDDDA
AU17620261(c)3L02,L03,L04write G matrix; find kerG and dim imGACDDDDBDDBDDCDDDDDDDCDDDA
AU17720261(d)4V03,L03,L12characteristic not equal to 2 prove when kerD∩imG=0ACDDDCCDDCDDCBDDDDDDCCDDA
AU17820261(e)2V03,V06,L04deduce from dimensions F5=kerD+imG and find rankDADDDDCCDDCDDCBDDDDDDDDDDA
AU17920261(f)2L12,V03characteristic 2 construct a nonzero intersection element whenACDDDCDDDCDDCCDDDDDDBBDDA
AU18020262(a)3V08,L01,L02prove J:p(x)↦p(-x) linear, and write its matrixCCDDDCBDDDDDDCDDDDDDADDDA
AU18120262(b)7V02,V04,L11,L12find the even / basis of the odd subspace, intersection and characteristic 2 classificationBCDDCCBBDCDDCDDDCDDDAADDA
AU18220262(c)2L02,V08write the differentiation-operator matrixCCDDDCCDDDDDDDDDDDDDCDDDC
AU18320262(d)4L09,L10prove DJ=-JD and deduce D interchange two subspacesDCDDDDCDDDDDDCDDDDDDCCDDC
AU18420262(e)4L10,L12,L04by characteristic 2/3 find the rank of the restricted mapBCDDDDBDDBDDCDDDDDDDAADDA
AU18520263(a)(i)3S01,S03find the eigenvalues of a symmetric matrixDDCDDDDCDDDDDDCDCDDDDCADA
AU18620263(a)(ii)4S03,S04construct an orthogonal Q and diagonal DDDADDDDDDDDDDDBDDDCDDDADA
AU18720263(a)(iii)3S07,S03construct B such that A=BB^TDDADDDDDDDDDDDCDDDDDDDADA
AU18820263(b)(i)3S11,M12,S05from a unitary Z=A+iB prove the real block matrix is orthogonalDDADDDDDCDCDDDDDDDCDDDDDA
AU18920263(b)(ii)3M12prove the block matrix and J commutativityDDCDDDDDBDCDDDDDDDDDDDDDB
AU19020263(b)(iii)4M12,S11,S05prove orthogonality and with J a commuting matrix must arise from a unitary matrixDDADDDDDBDCDDDDDDDCDDDDDA
AU19120264(a)(i)-(iii)8P01disjoint cycles, transposition factorisation and signDDDADDDDDDDDDDDDDDDDCDDCA
AU19220264(b)(i)3P04prove conjugacy in a group is an equivalence relationDDDADDDDDDDDDDDDDDDDDDDDA
AU19320264(b)(ii)2P04prove conjugation is compatible with positive integer powersDDDADDDDDDDDDDDDDDDDDDDDA
AU19420264(c)7P04,P02prove S_n two permutations are conjugate if and only if their cycle types agreeDDDADDDDDDDDDDDDDDDCDDDCA