Every quantitative portfolio, every factor model, every hedge is, underneath the finance vocabulary, a vector: a list of numbers combined by addition and scaling. A portfolio is a vector of weights; a factor exposure profile is a vector of betas; a hedge ratio is a scalar multiplying a vector of instrument holdings. This topic gives the language for that idea before any of the machinery — elimination, subspaces, eigenvalues — that the rest of the unit builds on it.
Three moves carry the whole topic. First, vectors add and scale, and every combination is a linear combination — the single most-used operation in the entire unit. Second, the dot product turns a pair of vectors into a number that measures alignment, giving length, angle, and the algebraic test for orthogonality. Third, a matrix acting on a vector, , is itself a linear combination — of the matrix's columns, weighted by 's entries. That last equivalence, the column picture, is the lens the whole unit views matrices through, from solving in the next topic to diagonalisation in Stage 3.
By the end of this topic you should be able to:
- form and geometrically interpret linear combinations of two or three vectors;
- compute the dot product, length, and angle between vectors, and test orthogonality;
- state and use the Cauchy–Schwarz and triangle inequalities;
- read both as a combination of columns and as a row of dot products, and explain why the two pictures agree.
In this lesson
- Contents
- The big picture
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