diff --git a/tech_docs/math_objects_python.md b/tech_docs/math_objects_python.md index c5432f4..1ad6476 100644 --- a/tech_docs/math_objects_python.md +++ b/tech_docs/math_objects_python.md @@ -1,27 +1,31 @@ # Mathematical Objects and Their Applications in Python -This document explains different types of mathematical objects and their applications. Additionally, it provides information on Python libraries that can be used to work with these objects. +This document explains different types of mathematical objects, their applications, and the specific datasets they are used with. It also provides information on Python libraries that can be used to work with these objects. ## Scalar -A **scalar** is a single number, typically used in mathematics to represent a magnitude or quantity. In programming and data science, scalars are often used as individual data points or constants. +A **scalar** is a single number, typically used in mathematics to represent a magnitude or quantity. In programming and data science, scalars are often used as individual data points, constants in equations, or as scale factors in algorithms. ### Example $$ a = 3 $$ +### Datasets +Scalars are used in virtually every dataset as individual measurements or metadata. For example, the age of a person in a demographic study is a scalar. ### Python Libraries - **NumPy**: Used for handling numerical operations; scalars can be represented as `numpy.float64` or `numpy.int32`, etc. ## Vector -A **vector** is an ordered array of numbers, which can represent direction and magnitude in space. Vectors are fundamental in physics for describing velocities and forces, in engineering for load distributions, and in machine learning for feature representation. +A **vector** is an ordered array of numbers, which can represent direction and magnitude in space. Vectors are fundamental in representing individual data points in feature space in machine learning, making them crucial for algorithms like support vector machines and in physics for describing velocities and forces. ### Example $$ \mathbf{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} $$ +### Datasets +Vectors are common in natural language processing where each word or document can be represented as a vector in a high-dimensional space (word embeddings). Also, in physics simulations, vectors represent forces, speeds, and directions. ### Python Libraries - **NumPy**: Provides a powerful array structure that can be used to represent vectors. @@ -29,12 +33,14 @@ $$ \mathbf{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} $$ ## Matrix -A **matrix** is a two-dimensional array of numbers used extensively in engineering, physics, computer science, and statistics. Matrices are crucial in solving systems of linear equations, transforming geometric data, and in algorithms for machine learning and artificial intelligence. +A **matrix** is a two-dimensional array of numbers used extensively to represent data transformations, relationships, or multivariate data. In statistics, matrices are used to describe data sets where rows might represent individual samples and columns represent features. ### Example $$ \mathbf{M} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$ +### Datasets +Matrices are central to image processing where each image can be represented as a matrix of pixel values. In social sciences, matrices can represent social networks where elements indicate the presence or strength of relationships. ### Python Libraries - **NumPy**: Essential for creating and manipulating matrices. @@ -43,7 +49,7 @@ $$ \mathbf{M} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$ ## Tensor -A **tensor** extends the concept of vectors and matrices to potentially higher dimensions. In the field of machine learning and neural networks, tensors are key structures, representing complex datasets and the parameters of the models themselves. +A **tensor** extends the concept of vectors and matrices to potentially higher dimensions, used in machine learning and neural networks to represent complex datasets and the parameters of the models themselves. ### Example @@ -54,6 +60,8 @@ $$ \end{bmatrix} $$ +### Datasets +Tensors are widely used in deep learning, especially in fields like computer vision where they represent images with different channels (RGB) in convolutional neural networks, or in natural language processing for embedding layers in models. ### Python Libraries - **TensorFlow**: A library designed specifically for working with tensors in neural networks. @@ -62,4 +70,4 @@ $$ ## Conclusion -Understanding these basic mathematical objects is crucial for working effectively in fields such as data science, AI, physics, and engineering. Python, with its rich ecosystem of libraries, provides extensive support for manipulating these objects, making it a preferred language for scientific and engineering applications. \ No newline at end of file +Understanding these basic mathematical objects and the datasets they relate to is crucial for working effectively in fields such as data science, AI, physics, and engineering. Python, with its rich ecosystem of libraries, provides extensive support for manipulating these objects, making it a preferred language for scientific and engineering applications.