QuantPrep | AI-Powered Learning for Quantitative Finance

We've assembled our toolkit. We have our fundamental objects (vectors) and the rules for manipulating them (addition and scalar multiplication).

Now, we can ask the most important creative question in all of linear algebra: With a given set of vectors, what can we build?

The answer to this question lies in two beautifully linked concepts: Linear Combinations and Span.

Linear Combinations: The Recipe

What is a Linear Combination?

A linear combination is the fundamental "recipe" for building new vectors out of old ones. It's an expression made from two ingredients: Scalar Multiplication (scaling your ingredient vectors) and Vector Addition (mixing them together).

Let's say we have two vectors, $v = [1, 2]$ and $w = [-3, 1]$ . A linear combination is any vector that can be written in the form:

c_1v_1 + c_2v_2 + \dots + c_nv_n

Where $c_1, c_2, \dots$ are any scalars you choose. Let's compute one:

2v + 1w = 2\begin{bmatrix} 1 \\ 2 \end{bmatrix} + 1\begin{bmatrix} -3 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 5 \end{bmatrix}

So, the vector $[-1, 5]$ is one possible linear combination of $v$ and $w$ .

Span: All the Things You Can Possibly Build

What is Span?

The set of all possible vectors you can create through a linear combination of a set of vectors is called the span of those vectors.

If we only had vector $v$ , its span would be the infinite line it sits on. But with both $v$ and $w$ , which point in different directions, we can reach any point in the entire 2D plane by choosing different scalars. The span of $v$ and $w$ is all of 2D space ( $\mathbb{R}^2$ ).

The Core Idea for Quants & ML

Span tells you the "expressive power" of your features. Imagine your model is `Stock Price = c₁ * (GDP Growth) + c₂ * (Interest Rate)`. The span of the "GDP" and "Interest Rate" vectors represents all possible outcomes your model can explain. If you add a new factor that is just a multiple of GDP growth, you haven't expanded the span at all. You've added redundant information.

Linear Combinations and Span

Linear Combinations: The Recipe

Span: All the Things You Can Possibly Build

The Core Idea for Quants & ML