A portfolio held across ten assets is a list of ten weights. A factor model's exposure profile is a list of betas. Neither object is "just a list" once you notice that portfolios add (combine two books into one) and scale (double a position, halve a hedge) — and that every operation a desk performs on a book of positions is built from exactly those two moves. This note fixes the vocabulary: vector addition, scalar multiplication, and their composite, the linear combination, together with the geometric fan-out — line, then plane, then all of space — that taking every combination of a growing set of vectors produces. That fan-out reappears constantly for the rest of the unit, from the column space of a matrix to the eigenspace of a matrix, so it earns careful treatment even though the individual operations are simple.
Learning objectives. By the end of this note you should be able to:
- add and scale vectors in , both algebraically (component by component) and geometrically (head-to-tail);
- form a linear combination and recognise the sum, difference, and zero vector as special cases;
- describe the set of all combinations of one, two, or three vectors as a line, a plane, or all of ;
- define the span of a set of vectors and identify when a new vector lies in it.
In this lesson
- Vectors and the two operations
- Linear combinations
- The geometric fan-out: line, plane, space
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