"Linear algebra is the branch of mathematics concerning linear equations..."
The study of systems of linear equations and their properties, including concepts like matrices, vectors, and eigenvectors.
Vectors and Matrices: These are essential concepts in Linear Algebra. Vectors refer to the quantity with both magnitude and direction, while matrices are grids of numbers that are used for representation and manipulation of data.
Linear Equations: Linear equations are essential in understanding the concepts of linear algebra. They are a set of mathematical relationships that are linear in terms of one or more variables.
Systems of Linear Equations: Multiple linear equations that need to be solved simultaneously are referred to as systems of linear equations. Solving such systems requires the use of matrix operations.
Determinants: A determinant is a scalar value used to describe a family of matrices, which has particular properties, such as invertibility, rank, and eigenvalues.
Matrix Algebra: Matrix algebra is the arithmetic operations and properties held by matrices that allow for the calculation of solutions to various algebraic problems.
Matrix Inverse: The inverse of a matrix is a matrix that can be multiplied by the original matrix to produce the identity matrix.
Eigenvalues and Eigenvectors: Eigenvectors are vectors that maintain their direction after being transformed, while eigenvalues are scalars used to represent the degree of such transformation.
Matrix Rank: Rank represents the number of linearly independent rows or columns in a matrix.
Matrix Transpose: In the transpose of a matrix, the rows of the original matrix become columns and vice versa.
Vector Spaces: Vector spaces are sets of vectors that satisfy specific properties under addition and scalar multiplication.
Linear Transformations: Linear transformations are operations that take in vectors and produce new ones in a way that preserves the linearity of the relationship between them.
Inner Products: Inner products are scalar values that describe the relationship between two vectors in a space.
Orthogonality: Orthogonality refers to the property of two vectors being perpendicular to each other.
Norms: Norms are functions that provide measures for the length or size of a vector.
Gram-Schmidt Process: This process is used to convert a set of vectors into an orthonormal set that spans the same subspace.
Singular Value Decomposition: This is a matrix operation that expresses any rectangular matrix as the product of three other matrices, each with specific properties.
Principal Component Analysis: PCA is a statistical technique used to reduce the dimensions of large datasets and identify the most important variables that contribute to the variations in the data.
Linear Regression: Linear regression refers to the statistical technique used to establish the relationship between two or more variables.
Multivariate Regression: When linear regression techniques are used on multiple independent variables, it is referred to as a multivariate regression model.
Time Series Analysis: Time series analysis is a statistical technique used to analyze temporal data, specifically to identify and model trends, seasonality, and other time-dependent structures.
Linear Regression Analysis: It is a statistical technique in which a linear relationship is established between the dependent and independent variables.
Multiple Regression Analysis: It is an extension of linear regression that involves more than one independent variable.
Ridge Regression: It is a form of linear regression that involves the use of a penalty term to prevent overfitting of the model.
Lasso Regression: It is another form of linear regression that involves the regularization of the model to reduce the number of variables used in the model.
Panel Data Analysis: It is a statistical technique used to analyze data sets that involve observations over time and across individuals.
Time Series Analysis: It is a statistical technique used to analyze data sets that involve observations of a single variable over time.
Factor Analysis: It is a statistical technique used to identify underlying factors in a dataset and reduce the number of variables used in a model.
Principal Component Analysis: It is a statistical technique used to reduce the complexity of a dataset by identifying the most important variables.
Canonical Correlation Analysis: It is a statistical technique used to identify the relationship between two sets of variables.
Cluster Analysis: It is a statistical technique used to group observations based on similarities in their characteristics.
"For instance, a₁x₁ + ... + aₙxₙ = b can be considered a linear equation."
"Linear maps such as (x₁, ..., xₙ) ↦ a₁x₁ + ... + aₙxₙ..."
"...their representations in vector spaces and through matrices."
"Linear algebra is central to almost all areas of mathematics."
"Linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes, and rotations."
"Functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions."
"Linear algebra is also used in most sciences and fields of engineering..."
"...because it allows modeling many natural phenomena..."
"...and computing efficiently with such models."
"For nonlinear systems, which cannot be modeled with linear algebra..."
"...it is often used for dealing with first-order approximations..."
"...using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point."
"For instance, a₁x₁ + ... + aₙxₙ = b can be considered a linear equation."
"Linear algebra is also used in most sciences and fields of engineering..."
"Linear algebra is also used in most sciences and fields of engineering..."
"Linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes, and rotations."
"Functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions."
"For nonlinear systems, which cannot be modeled with linear algebra..."
"...it is often used for dealing with first-order approximations..."