Vectors in Data Science

Study Notes

Definition: A vector is a mathematical object used to represent quantities with both magnitude and direction.

Key Concepts:

Scalar: A single value used to scale a vector.
Vector Operations:
- Addition: Combining two or more vectors by adding corresponding elements.
- Scalar Multiplication: Multiplying a vector by a scalar.
Vector Properties:
- Commutativity: The order of vectors in an operation does not affect the result.
- Associativity: The order in which vectors are combined does not affect the result.
- Distributivity: A scalar can be distributed across multiple vectors.

Definition: A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Key Concepts:

Matrix Operations:
- Matrix Addition: Combining two matrices by adding corresponding elements.
- Matrix Multiplication: Combining two matrices by multiplying rows of one matrix with columns of another.
Matrix Properties:
- Associativity: The order in which matrices are combined does not affect the result.
- Distributivity: A matrix can be distributed across multiple matrices.
Types of Matrices:
- Square Matrix: A matrix with an equal number of rows and columns.
- Identity Matrix: A square matrix with all elements on the main diagonal equal to 1 and all other elements equal to 0.
- Zero Matrix: A matrix with all elements equal to 0.

Vector Representation: Vectors are used to represent data points in high-dimensional spaces.
Matrix Representation: Matrices are used to represent relationships between data points, such as covariance and correlation matrices.
Linear Algebra Operations: Vector and matrix operations are used in machine learning algorithms, such as linear regression and neural networks.
Data Transformation: Matrices are used to transform data, such as scaling, rotating, and projecting data.

A vector is a mathematical object used to represent quantities with both magnitude and direction.
Scalars are single values used to scale vectors.
Vector operations include addition, which combines two or more vectors by adding corresponding elements, and scalar multiplication, which multiplies a vector by a scalar.
Vector properties include commutativity, where the order of vectors in an operation does not affect the result, associativity, where the order in which vectors are combined does not affect the result, and distributivity, where a scalar can be distributed across multiple vectors.

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
Matrix operations include matrix addition, which combines two matrices by adding corresponding elements, and matrix multiplication, which combines two matrices by multiplying rows of one matrix with columns of another.
Matrix properties include associativity, where the order in which matrices are combined does not affect the result, and distributivity, where a matrix can be distributed across multiple matrices.
Types of matrices include square matrices, which have an equal number of rows and columns, identity matrices, which are square matrices with all elements on the main diagonal equal to 1 and all other elements equal to 0, and zero matrices, which have all elements equal to 0.

Vectors are used to represent data points in high-dimensional spaces.
Matrices are used to represent relationships between data points, such as covariance and correlation matrices.
Vector and matrix operations are used in machine learning algorithms, such as linear regression and neural networks.
Matrices are used to transform data, such as scaling, rotating, and projecting data.