Linear Algebra
107 questions across 7 topics. Also examined in CNET, ISS, GATE ST, GATE DA, JAM MS.
Algebra of Matrices, Rank, Determinant, Linear Equations
- 13Matrix geometric seriesMSET June 2025Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 14Determinants — multiplicativity and scalingMSET June 2025Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 15Linear systems — consistencyMSET June 2025Algebra of Matrices, Rank, Determinant, Linear EquationsHard
- 19The adjoint (adjugate) matrixMSET June 2025Algebra of Matrices, Rank, Determinant, Linear EquationsHard
- 12Matrix rank — AB versus BA for rectangular A,BMSET April 2024Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 13Linear systems — consistency of an underdetermined systemMSET April 2024Algebra of Matrices, Rank, Determinant, Linear EquationsHard
- 14Matrix rank — Sylvester's inequalityMSET April 2024Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 17Matrices — singular values via AA^mathrm TMSET April 2024Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 18Similar matrices — everything polynomial is preservedMSET April 2024Algebra of Matrices, Rank, Determinant, Linear EquationsHard
- 19Nilpotent matrices — shifted invertibilityMSET April 2024Algebra of Matrices, Rank, Determinant, Linear EquationsHard
- 20Orthogonal matrices — forcing an eigenvalue of 1MSET April 2024Algebra of Matrices, Rank, Determinant, Linear EquationsHard
- 100Orthogonal matrix — the inverse is the transposeMSET April 2024Algebra of Matrices, Rank, Determinant, Linear EquationsHard
- 15Shifting by a scalar multiple of I preserves eigenvectorsMSET March 2023Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 16Determinant of a function of a matrix, via its eigenvaluesMSET March 2023Algebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 17What matching characteristic polynomials guarantee — and what they don'tMSET March 2023Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 28A real 3x3 matrix must have a real eigenvalue — ruling out a purely-complex minimal polynomialMSET September 2021Algebra of Matrices, Rank, Determinant, Linear EquationsHard
- 29A block matrix with a genuinely complex 2times2 blockMSET September 2021Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 30Same rank alone does not force congruence — Sylvester's law of inertia also needs the signatureMSET December 2020Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 79The first fundamental form is always non-singular; the second need not beMSET December 2020Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 81A local max and a local min of a graph function both give non-negative Gaussian curvatureMSET December 2020Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 29Nonzero eigenvalues (real or complex) already force invertibilityMSET June 2019Algebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 43Composing two Jacobians via the chain ruleMSET June 2019Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 79Gaussian curvature as the ratio of the two fundamental form determinantsMSET June 2019Algebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 5The 3times3 Fourier matrix squares to the index-reversal permutationMSET January 2018 Paper IIAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 7rank(AA^t)=rank A for real matricesMSET January 2018 Paper IIAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 43Multiplying by an invertible matrix preserves rankMSET January 2018 Paper IIAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 5det(cA)=c^ndet A with c=det AMSET April 2017 Paper IIAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 6(AA^t)^t=AA^tMSET April 2017 Paper IIAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 7Row 3 equals row 1 + row 2 on the left but not on the rightMSET April 2017 Paper IIAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 8A'A=I gives (det A)^2=1MSET April 2017 Paper IIAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
Vector Spaces, Subspaces, Linear Dependence, Basis, Dimension
- 11Linear algebra — real dimension of complex upper-triangular matricesMSET April 2024Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 64Linear independence on subintervals — ODE solutions are specialMSET April 2024Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionHard
- 66Wronskian of a triple root — Abel's formula, computed directlyMSET April 2024Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionHard
- 11Checking three sets for linear independence — and finding all three passMSET March 2023Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionHard
- 12Dimension of the singular locus inside the symmetric matricesMSET March 2023Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 60Gluing the endpoints of an interval produces a circleMSET March 2023Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 64Two functions with nonvanishing Wronskian can always be realised as ODE solutionsMSET March 2023Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 5Dimension of the space of polynomials of bounded degreeMSET September 2021Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 6Injective linear maps — what they preserve, and what they don't forceMSET September 2021Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionHard
- 27Dimension of the skew-symmetric matricesMSET September 2021Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 6The one false statement about an eigenvalue's defining kernelMSET December 2020Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 28The rank–nullity theorem, restated for the quotient spaceMSET December 2020Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 29Reading a linear operator's matrix off its coordinate formulasMSET December 2020Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 33A vanishing Wronskian that does not signal linear dependenceMSET December 2020Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 45Two exponentials, linearly independent, forced to vanish identicallyMSET December 2020Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 54The classic non-Noetherian ring: polynomials in infinitely many variablesMSET December 2020Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 5Doubling real dimension when a complex space is viewed over RMSET June 2019Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 6A basis-shift-down operator is nilpotent of index nMSET June 2019Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 26Detecting a hidden linear dependence among the spanning vectorsMSET June 2019Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 30The adjoint's matrix is the conjugate transposeMSET June 2019Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 33The solution space of an nth order linear ODE is exactly n-dimensionalMSET June 2019Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 6A skew-symmetric matrix is fixed by its entries above the diagonalMSET January 2018 Paper IIVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 20The trace is one linear condition on n^2 entriesMSET January 2018 Paper IIVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 41A linear equation defines a subspace; x1x2+x3=0 does notMSET April 2017 Paper IIVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
Eigenvalues and Eigenvectors, Cayley-Hamilton Theorem
- 16Eigenvalues of a polynomial in AMSET June 2025Eigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 18The all-ones matrixMSET June 2025Eigenvalues and Eigenvectors, Cayley-Hamilton TheoremHard
- 44Embedding finite groups in alternating groups; Sn is not nilpotentMSET April 2024Eigenvalues and Eigenvectors, Cayley-Hamilton TheoremHard
- 27A minimal polynomial that splits over R guarantees triangulability, not diagonalizabilityMSET December 2020Eigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 8Eigenvalue 1 has a two-dimensional eigenspace, so A is diagonalisableMSET January 2018 Paper IIEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 44lambda^2-(tr A)lambda+det A=lambda^2+1MSET January 2018 Paper IIEigenvalues and Eigenvectors, Cayley-Hamilton TheoremEasy
- 19A-I has rank 1, so the eigenvalue 1 needs no Jordan block of size 2MSET April 2017 Paper IIEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 39A square-free minimal polynomial means diagonalisable; trace 8, determinant 12MSET April 2017 Paper IIEigenvalues and Eigenvectors, Cayley-Hamilton TheoremEasy
- 43Diagonalisable means the eigenspace for 5 has full dimension 3MSET April 2017 Paper IIEigenvalues and Eigenvectors, Cayley-Hamilton TheoremEasy
- 4Block triangular: eigenvalues of the 2times2 corner plus the last diagonal entryMSET April 2017 Paper IIIEigenvalues and Eigenvectors, Cayley-Hamilton TheoremEasy
- 5The Perron eigenvector of a positive column-stochastic matrix is positiveMSET April 2017 Paper IIIEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
Algebra of Linear Transformations
- 12Linear maps — reconstructing T from its valuesMSET June 2025Algebra of Linear TransformationsHard
- 15Linear transformations — rank–nullity for a functionalMSET April 2024Algebra of Linear TransformationsEasy
- 16Linear transformations — properties of a projectionMSET April 2024Algebra of Linear TransformationsMedium
- 91Fredholm alternative — the two faces of the Fredholm determinantMSET April 2024Algebra of Linear TransformationsHard
- 13Recovering T(3,4) from linearity and two known valuesMSET March 2023Algebra of Linear TransformationsEasy
- 14AB=0 always means the column space of B sits inside the null space of AMSET March 2023Algebra of Linear TransformationsHard
- 49A linear map from R^2 compresses a compact square into a connected imageMSET March 2023Algebra of Linear TransformationsMedium
- 89Recognising a separable (degenerate) kernelMSET March 2023Algebra of Linear TransformationsEasy
- 5The matrix size of a linear map is codomain-by-domainMSET December 2020Algebra of Linear TransformationsEasy
- 26Spotting linearity: no constant shifts, no products of coordinatesMSET December 2020Algebra of Linear TransformationsMedium
- 74The inertia tensor connects two genuinely distinct physical vectorsMSET December 2020Algebra of Linear TransformationsHard
- 28Reading a linear map's matrix off its coordinate formulasMSET June 2019Algebra of Linear TransformationsEasy
- 39A cyclic vector does not make T nilpotent; it only rules out T=0MSET January 2018 Paper IIAlgebra of Linear TransformationsMedium
- 41A non-trivial kernel is an eigenspace for 0MSET January 2018 Paper IIAlgebra of Linear TransformationsEasy
- 4The difference quotient at 0 lowers each degree by one: a shift matrixMSET January 2018 Paper IIIAlgebra of Linear TransformationsMedium
- 14Express bi' in the basis b1,b2,b3 and use the coordinates as columnsMSET April 2017 Paper IIIAlgebra of Linear TransformationsMedium
Inner Product Spaces, Orthonormal Basis
- 19The one orthogonal-complement fact that needs no extra hypothesesMSET March 2023Inner Product Spaces, Orthonormal BasisMedium
- 30Orthogonal complement of a subspace defined by two linear equationsMSET September 2021Inner Product Spaces, Orthonormal BasisMedium
- 43A vector orthogonal to the gradientMSET December 2020Inner Product Spaces, Orthonormal BasisEasy
- 27Only the standard Hermitian form is positive definite on mathbf C^2MSET June 2019Inner Product Spaces, Orthonormal BasisEasy
- 80The Frenet formula bar t'=kappabar nMSET June 2019Inner Product Spaces, Orthonormal BasisEasy
- 5Orthonormal columns make M orthogonal — not symmetricMSET January 2018 Paper IIIInner Product Spaces, Orthonormal BasisEasy
- 33In a separable space an orthonormal set is countableMSET January 2018 Paper IIIInner Product Spaces, Orthonormal BasisMedium
- 34Every separable infinite-dimensional Hilbert space is isometric to ell^2MSET January 2018 Paper IIIInner Product Spaces, Orthonormal BasisEasy
- 44Parallelogram law: u+v^2+u-v^2=2u^2+2v^2MSET April 2017 Paper IIInner Product Spaces, Orthonormal BasisEasy
- 32Bessel's inequality: only countably many Fourier coefficients are non-zeroMSET April 2017 Paper IIIInner Product Spaces, Orthonormal BasisMedium
- 33A complete orthonormal set has no non-zero vector orthogonal to itMSET April 2017 Paper IIIInner Product Spaces, Orthonormal BasisMedium
- 34X=Moplus M^perp holds for every closed subspace of a Hilbert spaceMSET April 2017 Paper IIIInner Product Spaces, Orthonormal BasisMedium
Matrix Representation, Change of Basis, Canonical and Jordan Forms
- 17Nilpotent matricesMSET June 2025Matrix Representation, Change of Basis, Canonical and Jordan FormsHard
- 20Symmetric matrices are always diagonalizable — collapsing the Jordan form to the identityMSET March 2023Matrix Representation, Change of Basis, Canonical and Jordan FormsMedium
- 26Powers of a unipotent Jordan blockMSET September 2021Matrix Representation, Change of Basis, Canonical and Jordan FormsEasy
- 19The y-axis is fixed and the xz-plane turns through pi/4MSET January 2018 Paper IIMatrix Representation, Change of Basis, Canonical and Jordan FormsEasy
- 40Minimal polynomial = characteristic polynomial: one Jordan block per eigenvalueMSET January 2018 Paper IIMatrix Representation, Change of Basis, Canonical and Jordan FormsEasy
- 42Rotation by 90^circ swaps the two diagonal basis vectors up to signMSET January 2018 Paper IIMatrix Representation, Change of Basis, Canonical and Jordan FormsMedium
- 20A^3=0ne A^2 forces one nilpotent Jordan block of size 3MSET April 2017 Paper IIMatrix Representation, Change of Basis, Canonical and Jordan FormsEasy
- 40Exponent 2 on (x-1), exponent 1 on (x-2) in the minimal polynomialMSET April 2017 Paper IIMatrix Representation, Change of Basis, Canonical and Jordan FormsEasy
- 42The rows of the matrix are the coefficients of each outputMSET April 2017 Paper IIMatrix Representation, Change of Basis, Canonical and Jordan FormsEasy
Quadratic Forms, Reduction and Classification
- 20Quadratic form — definiteness by leading minorsMSET June 2025Quadratic Forms, Reduction and ClassificationMedium
- 158Definiteness of a quadratic formMSET June 2025Quadratic Forms, Reduction and ClassificationHard
- 158Checking positive definiteness of a quadratic form via leading minorsMSET March 2023Quadratic Forms, Reduction and ClassificationEasy
- 49The trace form for C/R reduces to 2Re(z^2)MSET June 2019Quadratic Forms, Reduction and ClassificationMedium
- 10A positive definite matrix need not be orthogonalMSET April 2017 Paper IIIQuadratic Forms, Reduction and ClassificationEasy