Brownian Bridge™

1. The rank-nullity theorem states that for $A \in \mathbb{R}^{m \times n}$ , $\operatorname{rank}(A) + \operatorname{nullity}(A)$ equals which of the following?

n (the number of columns of A)m (the number of rows of A)m + n (the total number of entries in each row and column)min(m, n) (the smaller dimension)

2. Let $A \in \mathbb{R}^{3 \times 3}$ with rows $(1, 2, 3)$ , $(4, 5, 6)$ , $(7, 8, 9)$ . What is $\operatorname{rank}(A)$ and $\operatorname{nullity}(A)$ ?

rank = 2, nullity = 1rank = 3, nullity = 0rank = 1, nullity = 2rank = 2, nullity = 0

3. The system $Ax = b$ with $A \in \mathbb{R}^{m \times n}$ has a unique solution if and only if which conditions hold?

b ∈ im(A) and ker(A) = {0}b ∈ im(A) and m = nker(A) = {0} and m ≥ nA is symmetric and b ∈ ℝⁿ

4. A sample covariance matrix $\Sigma = \frac{1}{T} X^\top X$ (where $X \in \mathbb{R}^{T \times n}$ has mean-zero columns) is always:

Symmetric and positive semi-definiteSymmetric and positive definiteSymmetric but not necessarily semi-definitePositive definite but not necessarily symmetric

5. For $A \in \mathbb{R}^{5 \times 8}$ with $\operatorname{rank}(A) = 4$ , what is the dimension of the left null space $\ker(A^\top)$ ?

1458

6. You are bootstrapping an interest rate curve with 6 instruments (3m, 6m, 1y, 2y, 5y, 10y swaps) and 6 unknown discount factors. The solver returns infinitely many solutions. Which statement is the correct diagnosis?

The 6×6 system matrix A is rank-deficient — at least two instruments are redundant or linearly dependentThe system is overdetermined — there are more equations than unknownsThe system matrix A is full rank but b is outside the column space of AThe discount factors must be negative, which is disallowed by the solver

7. To generate $n$ correlated standard normal samples with covariance $\Sigma$ (SPD), which factorisation of $\Sigma$ is the correct and numerically preferred approach?

Cholesky factorisation Σ = LLᵀ, then set z = Lε where ε ~ N(0, I)Eigendecomposition Σ = QΛQᵀ applied to the sample covariance, then multiply by QLU factorisation with partial pivoting, then apply the L factor to ε ~ N(0, I)Directly invert Σ and sample from N(0, Σ⁻¹) instead

8. A risk model uses portfolio variance $w^\top \Sigma w$ where $\Sigma \in \mathbb{R}^{100 \times 100}$ is a covariance matrix estimated from 40 daily returns. A portfolio manager asks: 'Can portfolio variance ever be exactly zero for a non-zero portfolio?' What is the correct answer?

Yes — with T = 40 < n = 100, rank(Σ) ≤ 40, so there exist non-zero w in the null space of Σ with wᵀΣw = 0No — covariance matrices are always positive definite, so wᵀΣw > 0 for all non-zero wYes — but only if the portfolio weights sum to zero (dollar-neutral constraint)No — wᵀΣw = 0 only when w = 0 by the definition of inner product

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