Vectors

A vector $\bm{x} \in \R ^n$ is an ordered list of $n$ numbers. $n$ is known as the dimensionality of $\bm{x}$.

$$ \bm{x} = \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} $$

Linear Combination

There are 2 vector operations which form the foundation of linear algebra (Strang).

1. Vector Addition: $\bm{z} = \bm{x} + \bm{y} \space \text{ where } \space \bm{x}, \space \bm{y}, \space \bm{z} \in \R^n$
   • Properties
2. Scalar Multiplication: $\alpha \bm{x} = \begin{bmatrix} \alpha x_1 & \dots & \alpha x_n\end{bmatrix}^T, \space \text{ for } \space \bm{x} \in \R^n, \alpha \in F$

These 2 operations establish the notion of a linear combination.

$$ \bm{z} = \sum _{k = 1} ^m \alpha _k \bm{x}_k $$

Parallelogram Law

For vectors $\bm{x}, \bm{y} \in \R ^n$, consider the parallelogram formed by graphically superimposing the vectors. Their sum $\bm{x} + \bm{y}$ is the main diagonal (the diagonal stemming from the tails of both vectors). Their differences $\bm{x} - \bm{y}$ and $\bm{y} - \bm{x}$ is the counter diagonal (the diagonal stemming from the tips of both vectors), where the tip of the resulting vector lies on the tip of the non-negated vector.

Examples of Vectors

Feature Vector

A vector $\bm{x}$ which encapsulates features of an object.

Portfolio Vector

A vector $\bm{x}$ with entries describing the investment of funds towards different financial instruments as a fraction of the total investment.