By default, vectors are visualized as column vectors, where their elements are stacked vertically. In formal mathematical notation, a vector $$\mathbf{x}$$ with elements $$x_1, \ldots, x_n$$ is represented as a column matrix enclosed in brackets:
$$ \mathbf{x} = \begin{bmatrix} x_{1} \\ \vdots \\ x_{n} \end{bmatrix} $$
This is in contrast to row vectors, whose elements are stacked horizontally.

Column Vector Representation

To refer to a specific element within a vector, mathematical notation uses a subscript attached to the variable name. For example, $$x_2$$ designates the second element of the vector $$\mathbf{x}$$. Because individual elements are scalars, they are written in standard text and are not bolded.

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The individual scalar values that comprise a vector are referred to as its elements. In mathematical and computational contexts, these are also interchangeably called entries or components.

Element of a Vector

In mathematics and deep learning, vectors are typically distinguished from scalars through specific typographical conventions. A standard approach is to denote a vector using a bold lowercase letter, such as $$\mathbf{x}$$, $$\mathbf{y}$$, or $$\mathbf{z}$$, which visually differentiates it from an unbolded scalar variable like $$x$$.

Typographical Conventions for Vectors

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Vector Element Subscript Notation

When vectors represent examples from real-world datasets, their scalar elements correspond to specific features or quantities with real-world significance. For instance, a vector representing a medical patient might have components corresponding to vital signs and cholesterol levels, while a vector for a loan applicant might encode income and employment history.

Real-World Significance of Vectors

A physics textbook contains the following statement: 'The momentum of an object, represented as **p**, is the product of its mass, *m*, and its velocity, **v**.' Based on the typographical conventions used in this statement, which of the quantities are vectors?

An equation is written as $\mathbf{x} = \mathbf{y} + z$. In the notation system being used, bold letters (like $\mathbf{x}$ and $\mathbf{y}$) represent quantities with multiple components, while italic letters (like $z$) represent quantities with a single component. A core principle of this system is that addition is only defined between quantities of the same type (i.e., multi-component can only be added to multi-component). Identify the typographical error in the equation and explain why it creates a logical inconsistency based on the given principle.

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