Unit 2 of 10
Linear algebra
The biggest unit. It grew to 10 questions in 2026, with Jordan forms and quadratic forms.
- 8–10questions expected next time
- 8asked in 2025
- 10asked in 2026
- 2ideas asked in both papers
Practise two-step problems: from a minimal polynomial to the Jordan form, then to a rank.
Most likely questions
Practise each one until you can do it without notes, then tick it. Ticks stay in this browser.
Already repeated
These came up in 2025 and again in 2026. Answer each in your head, then tap to check.
Rank and nullity of a given linear map
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rank + nullity = dimension of the domain.
Eigenvalues of a matrix, or from a polynomial that kills it
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Every eigenvalue is a root of any polynomial with p(A) = 0. A² = I gives ±1; A² = A gives 0 and 1.
Asked in 2026
10 questions. Tap one for the answer and working.
Q315-dim operator, min poly (x-2)^3(x-1): Jordan block structure, generalized eigenspace dim, rank of (T-2I)^2A
Answer: 3,3
Min poly (x-2)^3(x-1): gen. eigenspace of 2 has dim d=3 or 4. d=3 (one 3-block, eigenvalue 1 has mult 2): rank (T-2I)^2 = 1 + 2 = 3. d=4 gives (4,2), not an option; (4,1) impossible.
Sure
Q38Gram-Schmidt on (1,1,0),(1,0,1),(0,1,1); trace of projection onto first two vectorsA
Answer: 2
Projection onto a 2-dim subspace has trace = rank = 2.
Sure
Q75F,g independent in V*; T(x)=(f(x),g(x)): rank of T and dim kernelA
f,g independent -> T onto R^2, rank 2; nullity n-2
Sure
Q80Operator with rank(T)=2 and given eigenvalue data: find minimal polynomial, then trace and nullityA
T^2=T: min poly x^2-x; eigenvalues 1 (mult 2 = rank), 0 (mult 2). trace=2, nullity=4-2=2
Sure
Q83Hermitian A with A^2=I, 4-dim, equal multiplicities: eigenvalues, orthogonal decomposition, traceA
A^2=I, A Hermitian -> eigenvalues +-1; dim 4 with equal multiplicity (2 each) -> trace = 2-2 = 0
Sure
Q84Q=x^2+y^2+z^2+2xy+2yz+2zx: matrix form, diagonalise, rank and indexB
Q=(x+y+z)^2: matrix of all ones, eigenvalues 3,0,0 -> rank 1, positive index 1 -> (1,1)
Fairly sure
Rank 1. If "index" means the number of negative squares, the answer is A (1,0).
Q873x3 real symmetric with char poly (x-1)^2(x-3): diagonalizable?, quadratic form signatureC
Real symmetric -> diagonalizable; eigenvalues 1,1,3 all positive -> signature (3,0)
Sure
Q90Ax=b, A is 3x4 rank 2, consistent: dimension of solution space, null spaceC
A is 3x4 rank 2: nullity 4-2=2; solution set = particular + 2-parameter null space, dimension 2
Sure
Q93RCF has companion blocks of degrees 2 and 3: dimension, minimal and characteristic polynomialD
Companion blocks of degree 2 and 3 -> dimension 2+3=5
Sure
Q98A^3-3A^2+2A=0: eigenvalues, invertibility, rank of A(A-I)A
Char eq of 3x3 A: x(x-1)(x-2) so eigenvalues 0,1,2 (A singular, not invertible). A(A-I) has eigenvalues 0,0,2 -> rank 1
Sure
Asked in 2025
8 questions. Tap one for the answer and working.
Q46Nullity of T: V3→V2A
T(a1,a2,a3)=(a2,a3); kernel = {(a1,0,0)}, nullity 1.
Matches the official key
Q47Rank of given matrix AB
Row reduction (checked in python): rank 3.
Matches the official key
Q48Eigenvalues of given matrixB
Expand along row 2: (1-L)((1-L)^2-2)=0 => 1, 1+sqrt2, 1-sqrt2.
Matches the official key
Q49Dimension of subspace W1D
One independent linear constraint on R^5: dim 4.
Matches the official key
Q50Count of non-singular linear maps ℝ⁵→…A
Non-singular (invertible) maps need equal dimensions; R^5 -> R^3 cannot be, so 0.
Matches the official key
Q51Given T(e1),T(e2),T(e3): one-one/onto?D
T(e3)=T(e1)+T(e2), so T is singular: neither one-one nor onto.
Matches the official key
Q52Distance between orthogonal unit vectors (√2)B
|a-b|^2 = 1+1-2(0) = 2, distance sqrt2.
Matches the official key
Q53Rank and nullity of T: R²→R³D
Image spanned by (1,1,0),(1,-1,1): rank 2; nullity 2-2 = 0.
Matches the official key
In the syllabus, not asked yet
Give each of these one revision pass. Skipped topics often turn up later.
- Dual space and dual basis
- Change of basis
- Cayley–Hamilton, asked directly
- Unitary and skew-Hermitian properties
Full syllabus for this unit
Linear maps, kernel and range, rank–nullity, matrix of a map, change of basis, dual spaces, linear systems, eigenvalues, Cayley–Hamilton, diagonalisation, Hermitian and unitary matrices, inner products, Gram–Schmidt, quadratic forms, rational and Jordan forms.