Skip to content

Span

Span is the set of all linear combinations of given vectors. Test whether a target exposure can be reached, see when a new instrument adds coverage, and separate algebra from trading constraints.

The span of a collection of vectors is the set of every linear combination of those vectors. It answers a coverage question: can these building blocks produce the target I need?

For vectors v1,…,vk\mathbf v_1,\ldots,\mathbf v_k in a real vector space,

span⁡(v1,…,vk)={∑i=1kqivi:qi∈R}.\operatorname{span}(\mathbf v_1,\ldots,\mathbf v_k) =\left\{\sum_{i=1}^{k}q_i\mathbf v_i:q_i\in\mathbb R\right\}.
The original vectors belong to the span, but the span also includes their sums, fractional multiples, reversals and the zero vector. It is a set of possible results, not just the original list or one chosen combination. The real-number assumption matters: restricting the coefficients changes the attainable set. Axler, §2A, gives the formal definition and empty-list convention.

Two instruments, but only one direction of coverage

A hypothetical risk desk measures sensitivity to two specified factors, in a fixed order. Each entry is the change in portfolio value in USD per one percentage-point move of that factor, holding the other factor fixed. Assume instrument sensitivities are fixed at the current market state and add linearly across holdings.

One contract of instrument A has sensitivity vector a=(1,2)\mathbf a=(1,2); one contract of B has b=(2,4)\mathbf b=(2,4). Multiplying these per-contract vectors by contract quantities gives total sensitivity. We initially permit any real quantities, including fractional and short positions.

Every combination is

qAa+qBb=(qA+2qB,  2qA+4qB)=(t,2t),t=qA+2qB.q_A\mathbf a+q_B\mathbf b =(q_A+2q_B,\;2q_A+4q_B) =(t,2t),\qquad t=q_A+2q_B.
Because tt can be any real number, the span is the entire line through the origin satisfying y=2xy=2x. It extends beyond both arrows in both directions. Buying more of B cannot change the second-to-first sensitivity ratio.

Now compare two targets, in the same sensitivity units:

TargetIn the span?Reason
(3,6)(3,6)Yesa+b=3a=(3,6)\mathbf a+\mathbf b=3\mathbf a=(3,6)
(3,5)(3,5)NoEvery reachable vector has second component twice its first; 5≠2(3)5\ne2(3)

For the second target, matching factor 1 forces factor 2 sensitivity to be 6. Matching factor 2 at 5 forces factor 1 sensitivity to be 2.5. Rebalancing between A and B cannot remove that conflict. This is an exact failure of coverage, rather than a failure to search hard enough for holdings.

Hypothetical two-factor sensitivities in USD per percentage-point factor move. Vectors a=(1,2) and b=(2,4) reach (3,6) on y=2x but cannot reach (3,5). Adding c=(1,0) expands coverage to the plane and reaches (3,5) as 2.5a plus 0.5c.
Hypothetical two-factor sensitivities in USD per percentage-point factor move. Vectors a=(1,2) and b=(2,4) reach (3,6) on y=2x but cannot reach (3,5). Adding c=(1,0) expands coverage to the plane and reaches (3,5) as 2.5a plus 0.5c.
Original hypothetical fixed-sensitivity example. Each instrument vector is the exposure of one contract; targets are total exposures. The axes use equal scales. Shading shows only the visible part of an unbounded span; real contract quantities are allowed.

What actually expands a span?

A new vector expands coverage precisely when it lies outside the old span. Add instrument C with per-contract sensitivity c=(1,0)\mathbf c=(1,0). It breaks the relation y=2xy=2x. Indeed, every target (u,v)(u,v) has the construction
(u,v)=v2a+(u−v2)c.(u,v)=\frac v2\mathbf a+\left(u-\frac v2\right)\mathbf c.

So A and C span all of R2\mathbb R^2. In particular,

(3,5)=2.5a+0.5c.(3,5)=2.5\mathbf a+0.5\mathbf c.
B can remain available, but it adds no coverage beyond A. This is where span and linear dependence meet: span asks which targets exist, while dependence detects redundant generators. A basis covers the chosen space without redundancy. Here the original span has dimension 1; the expanded span has dimension 2.
For a matrix whose columns are instrument vectors, this same reachable set is its column space. Solving Aq=dA\mathbf q=\mathbf d tests whether the target d\mathbf d belongs to it. The MIT column picture makes that connection explicit.

Coverage has a boundary; execution has more boundaries

Being in the span is a statement about the specified linear model. It does not certify that a trade can be executed or funded.

  • Whole contracts: with integer quantities of A, B and C, the second component is 2qA+4qB2q_A+4q_B, always an even integer. The target (3,5)(3,5) is in the real span but remains impossible under this restriction.
  • Long-only positions: A, B and C have nonnegative components. Negative targets such as (−1,−2)(-1,-2) belong to their real span but require shorting, so they are unavailable with nonnegative quantities.
  • Funding and risk limits: the calculation has no price, margin, position-limit or liquidity condition. Matching two sensitivities also leaves other risks unspecified. Local sensitivity matching does not imply matching full cash flows after large market moves.

A fixed starting inventory is another distinction. If its exposure is z=(0,1)\mathbf z=(0,1) and allowed adjustments cover only the original line L={(t,2t):t∈R}L=\{(t,2t):t\in\mathbb R\}, final exposures fill

z+L={(t,1+2t):t∈R}.\mathbf z+L=\{(t,1+2t):t\in\mathbb R\}.
This is an affine translate, a shifted line satisfying y=2x+1y=2x+1, rather than a linear span. It does not contain the origin. To neutralise inventory, ask whether its negative is in the adjustment span; here −z=(0,−1)-\mathbf z=(0,-1) is outside LL.
Every span itself contains zero and is a subspace of its ambient vector space. Zero generators add no coverage; a collection consisting only of zero vectors, or the empty collection by convention, spans {0}\{\mathbf0\}. These edge cases reinforce the practical question: count independent directions of coverage, not instrument names.
For a geometric development, continue with Vectors, Linear Combinations, and Span. For the matrix solve behind a target-matching problem, use Vectors, Matrices and Linear Systems. Full lessons require membership; public previews introduce their scope.

Sources

  • Sheldon Axler, Linear Algebra Done Right, second edition, Chapter 2, “Span and Linear Independence,” p. 22.
  • Gilbert Strang, Introduction to Linear Algebra, fifth edition, §3.4, “Vectors that Span a Subspace,” pp. 167–168.
  • Sheldon Axler, Linear Algebra Done Right, fourth edition, §2A, definitions 2.2 and 2.4 and result 2.6.
  • MIT 18.06, Spring 2009 lecture summaries, Lectures 1, 6 and 7, on column combinations, column space and describing a complete solution subspace.
Last updated
  • linear algebra
  • target exposures
  • column space
  • hedging
Keep connecting the ideas
← Back to the glossaryBack to top ↑