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.

Analyzing Typographical Conventions

In the expression $\vec{F} = m\vec{a}$, the variable *m* is represented as a vector quantity.

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.

Vector Element Subscript Notation

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 $$x$$.

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Letters

A vector is an organism which transmits infection by conveying the pathogen from one host to another without causing disease itself, may be part of the infectious process.

Vector

Reference of Foundations of Large Language Models Course

Dive into Deep Learning

|$\KaTeX$ |`Plain Text`|
| ------- | ------- |
|$\Alpha$ | `\Alpha`|
|$\Beta$ | `\Beta`|
|$\Gamma$ | `\Gamma`|
|$\Delta$ | `\Delta`|
|$\Epsilon$ | `\Epsilon`|
|$\Zeta$ | `\Zeta`|
|$\Eta$ | `\Eta`|
|$\Theta$ | `\Theta`|
|$\Iota$ | `\Iota`|
|$\Kappa$ | `\Kappa`|
|$\Lambda$ | `\Lambda`|
|$\Mu$ | `\Mu`|
|$\Nu$ | `\Nu`|
|$\Xi$ | `\Xi`|
|$\Omicron$ | `\Omicron`|
|$\Pi$ | `\Pi`|
|$\Rho$ | `\Rho`|
|$\Sigma$ | `\Sigma`|
|$\Tau$ | `\Tau`|
|$\Upsilon$ | `\Upsilon`|
|$\Phi$ | `\Phi`|
|$\Chi$ | `\Chi`|
|$\Psi$ | `\Psi`|
|$\Omega$ | `\Omega`|
|$\varGamma$ | `\varGamma`|
|$\varDelta$ | `\varDelta`|
|$\varTheta$ | `\varTheta`|
|$\varLambda$ | `\varLambda`|
|$\varXi$ | `\varXi`|
|$\varPi$ | `\varPi`|
|$\varSigma$ | `\varSigma`|
|$\varUpsilon$ | `\varUpsilon`|
|$\varPhi$ | `\varPhi`|
|$\varPsi$ | `\varPsi`|
|$\varOmega$ | `\varOmega`|
|$\alpha$ | `\alpha`|
|$\beta$ | `\beta`|
|$\gamma$ | `\gamma`|
|$\delta$ | `\delta`|
|$\epsilon$ | `\epsilon`|
|$\zeta$ | `\zeta`|
|$\eta$ | `\eta`|
|$\theta$ | `\theta`|
|$\iota$ | `\iota`|
|$\kappa$ | `\kappa`|
|$\lambda$ | `\lambda`|
|$\mu$ | `\mu`|
|$\nu$ | `\nu`|
|$\xi$ | `\xi`|
|$\omicron$ | `\omicron`|
|$\pi$ | `\pi`|
|$\rho$ | `\rho`|
|$\sigma$ | `\sigma`|
|$\tau$ | `\tau`|
|$\upsilon$ | `\upsilon`|
|$\phi$ | `\phi`|
|$\chi$ | `\chi`|
|$\psi$ | `\psi`|
|$\omega$ | `\omega`|
|$\varepsilon$ | `\varepsilon`|
|$\varkappa$ | `\varkappa`|
|$\vartheta$ | `\vartheta`|
|$\thetasym$ | `\thetasym`|
|$\varpi$ | `\varpi`|
|$\varrho$ | `\varrho`|
|$\varsigma$ | `\varsigma`|
|$\varphi$ | `\varphi`|
|$\digamma$ | `\digamma`|

Greek Letters

|$\KaTeX$ |`Plain Text`|
| ------- | ------- |
|$\imath$ | `\imath`|
|$\nabla$ | `\nabla`|
|$\Im$ | `\Im`|
|$\Reals$ | `\Reals`|
|$\text{\OE}$ | `\text{\OE}`|
|$\jmath$ | `\jmath`|
|$\partial$ | `\partial`|
|$\image$ | `\image`|
|$\wp$ | `\wp`|
|$\text{\o}$ | `\text{\o}`|
|$\aleph$ | `\aleph`|
|$\Game$ | `\Game`|
|$\Bbbk$ | `\Bbbk`|
|$\weierp$ | `\weierp`|
|$\text{\O}$ | `\text{\O}`|
|$\alef$ | `\alef`|
|$\Finv$ | `\Finv`|
|$\N$ | `\N`|
|$\Z$ | `\Z`|
|$\text{\ss}$ | `\text{\ss}`|
|$\alefsym$ | `\alefsym`|
|$\cnums$ | `\cnums`|
|$\natnums$ | `\natnums`|
|$\text{\aa}$ | `\text{\aa}`|
|$\text{\i}$ | `\text{\i}`|
|$\beth$ | `\beth`|
|$\Complex$ | `\Complex`|
|$\R$ | `\R`|
|$\text{\AA}$ | `\text{\AA}`|
|$\text{\j}$ | `\text{\j}`|
|$\gimel$ | `\gimel`|
|$\ell$ | `\ell`|
|$\Re$ | `\Re`|
|$\text{\ae}$ | `\text{\ae}`|
|$\daleth$ | `\daleth`|
|$\hbar$ | `\hbar`|
|$\real$ | `\real`|
|$\text{\AE}$ | `\text{\AE}`|
|$\eth$ | `\eth`|
|$\hslash$ | `\hslash`|
|$\reals$ | `\reals`|
|$\text{\oe}$ | `\text{\oe}`|

Other Letters

The prime symbol (′) is a common mathematical notation used to denote a variable, such as `a′` or `v′`, that is derived from or related to another variable (`a` or `v`) without needing a new name. This notation, read as 'a-prime' or 'v-prime', can signify a value after a transformation, a derivative, or a subsequent state. It is frequently combined with subscripts, as in `v_i'`, which represents the prime version of the i-th element `v_i`.

Prime Notation in Mathematics

Typographical Conventions for Vectors

Vector borne disease is a disease that is transmitted to humans, plants, and animals by any arthropod species. 

Vector Borne Diseases

Vector competence defines the ability of a particular arthropod to transmit a given virus. For example, the overall mosquito barriers and antiviral responses toward pathogens would contribute to the mosquito's vector competence.

Vector Competence

Vectorial capacity describes the basic reproductive rate of a vector borne pathogen by a particular vector species or population. The formula below describes the total number of future infectious bites from mosquitos biting an individual infectious host on a single day.

Vectorial Capacity

Climate is a important factor that determines the rate of transmission of a disease. Changes in climate can lead to:
- Geographic shifts of vector and reservoirs
- Changes in rates of development, survival, reproduction of vectors, reservoirs, and pathogens

Such changes can lead to an increase in transmission and expansion of diseases in to other regions.


Climate and Vector Borne Diseases

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

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

To prevent ambiguity, deep learning terminology makes a strict distinction between 'order' and 'dimensionality'. The 'order' of a tensor refers to its number of axes, whereas 'dimensionality' exclusively refers to the number of components or elements along a specific axis (e.g., a vector's length).

Tensor Order vs. Dimensionality

To formally indicate that a vector $$\mathbf{x}$$ contains $$n$$ elements, it is written as $$\mathbf{x} \in \mathbb{R}^n$$. In this notation, $$n$$ defines the dimensionality of the vector, signifying that it belongs to the mathematical space of all $$n$$-dimensional vectors with real-valued components.

Dimensionality Notation for Real-Valued Vectors

A vector, denoted by a bold lowercase letter such as $$\mathbf{x}$$, can be defined by explicitly listing its elements. This component-wise representation is an ordered sequence of scalars, commonly written as $$\mathbf{x} = (x_1, x_2, \dots, x_n)$$, where $$n$$ represents the vector's dimensionality.

Component-wise Notation for a Vector

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.

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