1.1 Scalars and Vectors

: 30 minutes

This subsection defines the mathematical concepts behind vectors, matrices, tensors, and their codding counterparts.

Scalars

A scalar is a single (real) value, denoted by ordinary lower-cased letters (e.g., x, y, and z) and the space of all (continuous) real-valued scalars by \mathbb{R}. Consequently, if x is a real-valued scalar, we use the notation x\in\mathbb{R}.

Scalars are represented in Python by an int or float. For example, let x represent my weight in pounds.

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Python Scalars

Python offers four basic data types: int, float, bool, and str.

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Note Python Equality

What is the Python output from 0.1 + 0.1 + 0.1 == 0.3?

TrueFalse

This is because the face value 0.1 is represented by Python as a much longer floating number. Therefore, their sum does not match the float representation of the face value 0.3.

Vectors

In real-world applications, it oftentimes make more sense to group scalars together as a vector than letting them float around as seemingly unrelated variables.

For example, in a hypothetical clinical study of BMI, scalar measurements—like weight, height, and age—of an individual respondent can be put into to a feature vector for easer data manipulation.

A vector is a fixed-length array of scalars, called the elements of the vector. We denote vectors by bold lowercase letters, e.g., \bold{x}, \bold{y}, \bold{z}, etc.

The standard way to visualize a vector is by vertically stacking its elements:

\bold{x} =\begin{bmatrix}x_{1} \\ \vdots \\x_{n}\end{bmatrix}. \tag{1.1}

We can refer to an element of a vector by using a subscript. For example, x_2 denotes the second element of \mathbf{x}. Since x_2 is a scalar, we do not bold it.

If a vector \bold{x} contains n elements, we use the notation \bold{x}\in\mathbb{R}^n. Here, n is called the dimension or size of the vector.

TipRow vs Column Vectors

In linear algebra, we sometimes distinguish between such column vectors and row vectors whose elements are stacked horizontally. A row vector is written as follows: \bold{x} =\begin{bmatrix}x_{1}, \ldots, x_{n}\end{bmatrix}.

Note US States

Let \bold{x} denote the vector containing the number of counties in each of the states in the US. Which of the following must be true.

\bold{x}\in\mathbb{R}^{50}\bold{x}\in\mathbb{R}\bold{x}\in\mathbb{R}^{25}None of the above

There are only 50 states in the USA.

Vectors in Python

In Python, the code counterpart of a vector is simply a list.

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Warning0- vs 1-based Indexing

In Python, as in most programming languages, vector indices start at 0, also known as zero-based indexing, whereas in linear algebra subscripts begin at 1 (one-based indexing).