Vector Spaces

The Geometry of What You Can Build

A vector is a list of numbers — a portfolio's asset weights, a product's resource requirements, a week of daily sales. A vector space is the world of everything you can build from vectors using two moves: adding them and scaling them.

Linear combination — the building move:

$$w = \lambda_1 v_1 + \lambda_2 v_2 + \dots + \lambda_k v_k$$

Mix a bond portfolio and a stock portfolio in any proportions: every mix is a linear combination. The set of all combinations of some vectors is their span — the entire territory reachable from them.

Linear independence — the no-redundancy test. Vectors are independent if none of them is a combination of the others; equivalently, the only combination giving the zero vector is the all-zero one. Dependent vectors carry redundant information: if fund C is always 50/50 of funds A and B, adding C to your menu spans nothing new.

Basis = a set that is both independent (no waste) and spanning (no gaps): a minimal coordinate system. Every vector of the space is then a unique combination of basis vectors.

Dimension = the number of vectors in any basis — the space's count of genuine degrees of freedom:

Key idea: Dimension counts independent directions, not listed products.

A catalog of 40 mutual funds may live in a 5-dimensional space: five real bets dressed up forty ways. Diversifying across dependent assets is buying the same risk in different wrappers — the most expensive way to feel safe.

Connection backwards: $n$ vectors in $\mathbb{R}^n$ are independent exactly when the matrix stacking them has $\det \ne 0$. Determinants, rank, and dimension are three languages for one question: how much genuine variety is here?

Tip: To test independence in practice, stack the vectors as columns and compute the rank: rank = number of vectors means independent; a smaller rank tells you exactly how many are redundant.

Common pitfall: Confusing the number of vectors you listed with the dimension of what they span. Three portfolios built from the same two funds still span only a two-dimensional space — dimension counts independent directions, not products in the catalogue.