Scalars, Vectors, and Dimensions in Linear Algebra

A scalar is a single number. Examples include $7$, $-1.5$, and $0$.

A vector is an ordered collection of numbers. It is often written as a column:

$$\mathbf{x}=\begin{pmatrix}3\\2\end{pmatrix}.$$

The entries of a vector are called its components. The order matters:

$$\begin{pmatrix}3\\2\end{pmatrix}\ne\begin{pmatrix}2\\3\end{pmatrix}.$$

In geometry, a vector can represent an arrow. In data science, it can represent one item using several numerical features.

The dimension of a vector is the number of entries it contains.

• $\begin{pmatrix}4\end{pmatrix}$ has dimension $1$.
• $\begin{pmatrix}4\\-1\end{pmatrix}$ has dimension $2$.
• $\begin{pmatrix}4\\-1\\6\end{pmatrix}$ has dimension $3$.

A vector with $n$ real-number components belongs to $\mathbb{R}^n$:

$$\mathbf{v}\in\mathbb{R}^n.$$

For example,

$$\mathbf{v}=\begin{pmatrix}1.2\\0\\-5\\8\end{pmatrix}\in\mathbb{R}^4.$$

In machine learning, a vector commonly stores the features of one observation. For instance,

$$\mathbf{x}=\begin{pmatrix}12\\4\\0.8\end{pmatrix}$$

could represent three measurements, where each position has a fixed meaning:

The vector has dimension $3$ because there are three features. Every vector in a dataset must use the same feature order so that corresponding components can be compared correctly.

Worked example: Classify the object below:

$$\mathbf{a}=\begin{pmatrix}-2\\5\\1\end{pmatrix}.$$

1. It has three entries: $-2$, $5$, and $1$.
2. Therefore, it is a vector, not a scalar.
3. Its dimension is $3$.
4. In notation, $\mathbf{a}\in\mathbb{R}^3$.

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