Brownian Bridge™

Setup

Why linear algebra in quant finance?

Three operations dominate production quant code:

Portfolio aggregation: multiply a weight vector $w \in \mathbb{R}^n$ by a covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ to get portfolio variance $w^\top \Sigma w$ .
System solving: bootstrap a yield curve by solving a linear system $Ax = b$ where $A$ encodes instrument sensitivities to discount factors.
Factor decomposition: decompose a risk matrix to identify which linear combinations of assets drive most of the variance.

All three reduce to linear algebra. This module builds the precise foundations: what a vector space is, what a matrix does geometrically, and what the rank-nullity theorem tells you about the structure of solutions to linear systems.

INSIGHT

Why this matters on a derivatives desk. When bootstrapping a swap curve, the system $Ax = b$ has $A \in \mathbb{R}^{n \times n}$ encoding instrument-to-discount-factor relationships and $b$ containing observed par swap rates. If two instruments are economically equivalent (redundant), $A$ is singular. The rank-nullity theorem tells you exactly by how many degrees the solution is underdetermined — and therefore how many regularisation constraints you need to add to recover uniqueness.

Conventions

All vectors are column vectors in $\mathbb{R}^n$ unless stated otherwise.
$\mathbb{R}^{m \times n}$ denotes the set of real $m \times n$ matrices.
The standard inner product on $\mathbb{R}^n$ is $\langle x, y \rangle = x^\top y = \sum_{i=1}^n x_i y_i$ .
Notation: $\ker(A) = \{x \in \mathbb{R}^n : Ax = 0\}$ (null space); $\operatorname{im}(A) = \{Ax : x \in \mathbb{R}^n\}$ (column space / image).
Superscript $\top$ denotes transpose: $(A^\top)_{ij} = A_{ji}$ .

Theory

1. Vector Spaces

DEFINITION

Definition 1.1 (Vector Space). A vector space over $\mathbb{R}$ is a non-empty set $V$ equipped with two operations — addition $V \times V \to V$ and scalar multiplication $\mathbb{R} \times V \to V$ — satisfying eight axioms:

(Closure) $u + v \in V$ and $\alpha v \in V$ for all $u, v \in V$ , $\alpha \in \mathbb{R}$ .
(Commutativity) $u + v = v + u$ .
(Associativity) $(u + v) + w = u + (v + w)$ and $(\alpha\beta)v = \alpha(\beta v)$ .
(Zero vector) There exists $\mathbf{0} \in V$ with $v + \mathbf{0} = v$ for all $v$ .
(Additive inverses) For each $v \in V$ there exists $-v \in V$ with $v + (-v) = \mathbf{0}$ .
(Unit scalar) $1 \cdot v = v$ .
(Scalar distributivity) $\alpha(u + v) = \alpha u + \alpha v$ .
(Vector distributivity) $(\alpha + \beta)v = \alpha v + \beta v$ .

The canonical example is $\mathbb{R}^n$ : column vectors with componentwise addition and scaling. Other examples relevant to quant finance:

$\mathbb{R}^{m \times n}$ : the space of $m \times n$ matrices, used as parameter spaces in calibration.
$C[0, T]$ : continuous functions on $[0, T]$ , the natural home for asset price paths.
$L^2(\Omega, \mathcal{F}, \mathbb{P})$ : square-integrable random variables; the space in which conditional expectations and martingales live (covered in the Probability Theory course).

DEFINITION

Definition 1.2 (Subspace). A non-empty subset $W \subseteq V$ is a subspace if it is closed under addition and scalar multiplication — equivalently, if $\alpha u + \beta v \in W$ for all $u, v \in W$ and $\alpha, \beta \in \mathbb{R}$ .

The subspace test: $W$ is a subspace iff (i) $\mathbf{0} \in W$ and (ii) $W$ is closed under linear combinations.

EXAMPLE

Example 1.3 (Dollar-neutral portfolios). Let $V = \mathbb{R}^n$ be the space of portfolio weight vectors. The set $W = \{w \in \mathbb{R}^n : \mathbf{1}^\top w = 0\}$ of dollar-neutral portfolios (zero net exposure) is a subspace of dimension $n - 1$ . The sum of two dollar-neutral portfolios remains dollar-neutral; scaling preserves the constraint. This subspace arises in statistical arbitrage and market-neutral strategies.

2. Linear Spans and Bases

DEFINITION

Definition 2.1 (Linear span). The span of vectors $v_1, \ldots, v_k \in V$ is $\operatorname{span}(v_1, \ldots, v_k) = \left\{ \sum_{i=1}^k \alpha_i v_i : \alpha_i \in \mathbb{R} \right\}.$ This is the smallest subspace of $V$ containing all $v_i$ .

DEFINITION

Definition 2.2 (Basis and dimension). A basis of $V$ is a set $\{e_1, \ldots, e_n\}$ that is simultaneously:

Linearly independent: $\sum_i \alpha_i e_i = \mathbf{0}$ implies $\alpha_i = 0$ for all $i$ .
Spanning: $\operatorname{span}(e_1, \ldots, e_n) = V$ .

The number of elements in any basis is the dimension $\dim V$ — this is well-defined (any two bases of a finite-dimensional space have the same cardinality, proved via the Steinitz exchange lemma).

REMARK

Coordinate representation. Once a basis $\{e_1, \ldots, e_n\}$ is fixed, every $v \in V$ has a unique representation $v = \sum_i \alpha_i e_i$ . The coefficients $(\alpha_1, \ldots, \alpha_n)$ are the coordinates of $v$ in that basis. A change of basis is itself a linear map (an invertible matrix).

3. Matrices as Linear Maps

A matrix $A \in \mathbb{R}^{m \times n}$ defines a linear map $T_A : \mathbb{R}^n \to \mathbb{R}^m$ by $T_A(x) = Ax$ . Linearity is immediate: $T_A(\alpha x + \beta y) = A(\alpha x + \beta y) = \alpha Ax + \beta Ay = \alpha T_A(x) + \beta T_A(y).$

Conversely, every linear map between finite-dimensional spaces has a unique matrix representation once bases are fixed — the $j$ -th column of $A$ is the image of the $j$ -th basis vector of the domain.

DEFINITION

Definition 3.1 (Kernel and image). For $A \in \mathbb{R}^{m \times n}$ :

The null space (kernel): $\ker(A) = \{x \in \mathbb{R}^n : Ax = 0\}$ — a subspace of $\mathbb{R}^n$ .
The column space (image): $\operatorname{im}(A) = \{Ax : x \in \mathbb{R}^n\} = \operatorname{span}(\text{columns of } A)$ — a subspace of $\mathbb{R}^m$ .

Solvability: the system $Ax = b$ has a solution iff $b \in \operatorname{im}(A)$ . It has a unique solution iff additionally $\ker(A) = \{0\}$ .

THEOREM

Theorem 3.2 (Rank-Nullity). For any $A \in \mathbb{R}^{m \times n}$ : $\underbrace{\dim \operatorname{im}(A)}_{\operatorname{rank}(A)} + \underbrace{\dim \ker(A)}_{\operatorname{nullity}(A)} = n.$

Proof sketch. Let $r = \operatorname{rank}(A)$ . Choose a basis $\{u_1, \ldots, u_{n-r}\}$ for $\ker(A)$ , then extend to a basis $\{u_1, \ldots, u_{n-r}, v_1, \ldots, v_r\}$ of $\mathbb{R}^n$ . For any $x = \sum \beta_i u_i + \sum \gamma_j v_j$ , we have $Ax = \sum \gamma_j Av_j$ . It follows that $\{Av_1, \ldots, Av_r\}$ spans $\operatorname{im}(A)$ and is linearly independent (if a combination were zero, it would imply a dependence contradicting the extended basis). So $\dim \operatorname{im}(A) = r$ , giving $r + (n - r) = n$ . $\square$

EXAMPLE

Example 3.3 (Redundant swap instruments). Suppose five market instruments each price as a linear combination of six discount factors. The pricing equations are $Ax = b$ with $A \in \mathbb{R}^{5 \times 6}$ . By rank-nullity, $\operatorname{rank}(A) + \operatorname{nullity}(A) = 6$ . Since $A$ has only 5 rows, $\operatorname{rank}(A) \leq 5$ , so $\operatorname{nullity}(A) \geq 1$ : there is at least a one-dimensional family of solutions. The solution is not unique — additional constraints (e.g., smoothness of the forward curve) are required to select one.

4. The Four Fundamental Subspaces

For $A \in \mathbb{R}^{m \times n}$ with $\operatorname{rank}(A) = r$ , there are four canonical subspaces:

Subspace	Lives in	Dimension
Column space $\operatorname{im}(A)$	$\mathbb{R}^m$	$r$
Left null space $\ker(A^\top)$	$\mathbb{R}^m$	$m - r$
Row space $\operatorname{im}(A^\top)$	$\mathbb{R}^n$	$r$
Null space $\ker(A)$	$\mathbb{R}^n$	$n - r$

THEOREM

Theorem 4.1 (Orthogonality of fundamental subspaces). $\operatorname{im}(A) \perp \ker(A^\top) \quad \text{and} \quad \operatorname{im}(A^\top) \perp \ker(A).$ Furthermore, $\operatorname{im}(A) \oplus \ker(A^\top) = \mathbb{R}^m$ and $\operatorname{im}(A^\top) \oplus \ker(A) = \mathbb{R}^n$ (orthogonal direct sum decompositions).

Proof. Let $y \in \ker(A^\top)$ and $b = Ax \in \operatorname{im}(A)$ . Then $\langle b, y \rangle = (Ax)^\top y = x^\top A^\top y = x^\top \mathbf{0} = 0$ . The direct sum decomposition follows from dimension counting and orthogonality. $\square$

REMARK

Geometric picture. Think of $A$ as a map from $\mathbb{R}^n$ to $\mathbb{R}^m$ . The row space $\operatorname{im}(A^\top)$ is the "effective" part of $\mathbb{R}^n$ — the directions $A$ actually acts on. The null space $\ker(A)$ is the "invisible" part — directions $A$ crushes to zero. In $\mathbb{R}^m$ , the column space $\operatorname{im}(A)$ is the range and the left null space $\ker(A^\top)$ is the "unachievable" part.

5. Symmetric and Positive (Semi-)Definite Matrices

DEFINITION

Definition 5.1. A matrix $A \in \mathbb{R}^{n \times n}$ is:

Symmetric if $A^\top = A$ .
Positive semi-definite (SPSD) if $A^\top = A$ and $x^\top A x \geq 0$ for all $x \in \mathbb{R}^n$ .
Positive definite (SPD) if $A^\top = A$ and $x^\top A x > 0$ for all $x \neq 0$ .

Notation: $A \succeq 0$ (SPSD), $A \succ 0$ (SPD).

The quadratic form $x^\top A x$ has a geometric interpretation: it measures how much $A$ stretches $x$ in the direction of $x$ itself. Positive definiteness says $A$ never "collapses" any direction to zero or below.

THEOREM

Proposition 5.2 (Covariance matrices are SPSD). Let $X \in \mathbb{R}^{T \times n}$ be a returns matrix ( $T$ observations, $n$ assets) with mean-zero columns. The sample covariance matrix $\Sigma = \frac{1}{T} X^\top X \in \mathbb{R}^{n \times n}$ satisfies $\Sigma \succeq 0$ . It is positive definite iff $X$ has full column rank (no asset is a perfect linear combination of others).

Proof. $\Sigma^\top = (X^\top X)^\top = X^\top X = \Sigma$ (symmetric). For any $w \in \mathbb{R}^n$ : $w^\top \Sigma w = \frac{1}{T} w^\top X^\top X w = \frac{1}{T} \|Xw\|^2 \geq 0.$ The form equals zero iff $Xw = 0$ , i.e., $w \in \ker(X)$ . Full column rank of $X$ means $\ker(X) = \{0\}$ , giving positive definiteness. $\square$

WARNING

Near-singular covariance matrices. In practice, assets are highly correlated (e.g., equity index options at adjacent strikes), and $\Sigma$ estimated from finite data is frequently near-singular: it is theoretically SPD but has eigenvalues close to zero. Inverting $\Sigma$ directly (as required for minimum-variance weights $w^* \propto \Sigma^{-1}\mu$ ) amplifies these small eigenvalues and produces wildly unstable portfolio weights. The fix — regularisation via SVD or adding a diagonal ridge — is treated in Module 4.

Validation

The companion notebook verifies:

Vector space axioms — checks all eight axioms for $\mathbb{R}^3$ with explicit counterexamples showing what fails when each axiom is dropped.
Subspace test — confirms the dollar-neutral subspace is closed under linear combinations.
Rank-nullity — for a $4 \times 6$ matrix, computes $\operatorname{rank}(A)$ and $\operatorname{nullity}(A)$ via row reduction and verifies their sum equals 6.
Orthogonality of fundamental subspaces — constructs a basis for each of the four subspaces and verifies the pairwise orthogonality via dot products.
Covariance SPSD — builds a sample covariance from synthetic returns and confirms $w^\top \Sigma w \geq 0$ for 1000 random weight vectors.

PRACTICE

By hand before opening the notebook. Let $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}.$

Are the rows of $A$ linearly independent? What is $\operatorname{rank}(A)$ ?
By the rank-nullity theorem, what is $\operatorname{nullity}(A)$ ?
What is the dimension of $\ker(A^\top)$ (left null space)?
Does the system $Ax = (1, 2, 3)^\top$ have a solution? Does $Ax = (1, 1, 1)^\top$ ?

(Answers: $\operatorname{rank}(A) = 2$ since row 3 = 2 × row 2 − row 1; nullity = 1; $\dim \ker(A^\top) = 1$ ; $(1,2,3)^\top$ is column 1 of $A$ so yes; check whether $(1,1,1)^\top$ is in the column space.)

Limitations

WARNING

Numerical rank is not algebraic rank. The definitions above are exact, but in floating-point arithmetic, a matrix that is "theoretically" rank-2 may appear rank-3 due to rounding errors — or vice versa. Production code determines rank using singular values with a tolerance: count the number of singular values exceeding $\varepsilon \cdot \sigma_{\max}(A)$ for a problem-specific threshold $\varepsilon$ . This is treated rigorously in Module 4 (SVD and condition numbers).

WARNING

Invertibility is not conditioning. A matrix can be theoretically invertible but numerically catastrophic if its columns are nearly linearly dependent. The condition number $\kappa(A) = \|A\| \cdot \|A^{-1}\|$ quantifies this. A yield curve system with $\kappa \approx 10^{12}$ means computed solutions are meaningless below 12 significant digits — which is the entire double-precision range on a bad day. Always check condition numbers before trusting a linear solve in calibration.

Scope limitations:

This module assumes finite-dimensional spaces over $\mathbb{R}$ . Infinite-dimensional Hilbert spaces ( $L^2$ , Sobolev spaces) require additional structure (completeness, bounded operators) and arise in the functional analytic foundations of interest rate models.
Results extend to complex vector spaces (replacing transpose with conjugate transpose $A^*$ ), which are needed for Fourier-transform pricing (Module in fourier-fft-pricing course).

Vectors, Matrices, and Linear Maps

Setup

Why linear algebra in quant finance?

Conventions

Theory

1. Vector Spaces

2. Linear Spans and Bases

3. Matrices as Linear Maps

4. The Four Fundamental Subspaces

5. Symmetric and Positive (Semi-)Definite Matrices

Validation

Limitations

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