"Their use is also known as 'numerical integration', although this term can also refer to the computation of integrals."
Linear algebra is used extensively in computational physics, particularly in numerical methods for solving differential equations.
Vectors and Vector spaces: Fundamental objects of linear algebra on which all other concepts are based.
Linear transformations and matrices: The relationship between vectors and their transformations and how they are represented with matrices.
Systems of linear equations: Methods of finding solutions of systems of linear equations.
Determinants: How to calculate the determinant of a matrix and its properties.
Eigenvalues and eigenvectors: How to find eigenvectors and eigenvalues of a matrix.
Orthogonality and inner products: The concept of orthogonality and inner products in vector spaces.
Diagonalization and similarity transform: The diagonalization of matrices via similarity transformation.
Singular value decomposition: The decomposition of a matrix as a product of orthogonal and diagonal matrices.
Linear algebraic C++ libraries: Popular libraries for performing linear algebra computations and their capabilities.
Applications in Computational Physics: How linear algebra concepts can be used to solve real-world problems in physics, such as solving differential equations and modeling systems.
Matrix algebra: This type of linear algebra is concerned with the study of matrices, vectors, and their operations such as addition, subtraction, multiplication, and inversion.
Vector space: A vector space is a collection of vectors that satisfy certain properties, such as closure under vector addition and scalar multiplication. Vector spaces can be used to study linear transformations and eigenvalues/eigenvectors.
Linear transformations: Linear transformations are functions that take vectors from one vector space to another, while preserving their underlying structure. These transformations can be described using matrices or linear operators.
Systems of linear equations: This type of linear algebra deals with the study of systems of linear equations and their solutions. These equations can be solved using numerical algorithms, such as Gaussian elimination.
Eigenvalues and eigenvectors: Eigenvalues and eigenvectors are important concepts in linear algebra, used to study the behavior of linear transformations. Eigenvalues represent scalar values that are associated with a particular linear transformation, while eigenvectors represent vectors that are transformed only by a scalar multiple.
Singular value decomposition: Singular value decomposition (SVD) is a technique used to represent a matrix as a product of three matrices, with the middle matrix containing diagonal entries called singular values. SVD can be used for data compression, signal processing, and image processing.
Matrix factorization: Matrix factorization is a technique used to factor a matrix into two or more matrices, which can be used to approximate the original matrix. Matrix factorization is commonly used in data analysis, collaborative filtering, and recommendation systems.
Multilinear algebra: Multilinear algebra is a generalization of linear algebra that deals with multiple vector spaces and the operations between them. Multilinear algebra is used in theoretical physics, chemistry, and computer science.
Convex optimization: Convex optimization is a subfield of linear algebra that deals with finding the optimal solution to a convex optimization problem. Convex optimization is used in machine learning, engineering, and operations research.
Numerical linear algebra: Numerical linear algebra refers to the algorithms and methods used to solve linear algebra problems using numerical methods, such as the iterative methods, eigenvalue algorithms, and matrix decompositions. Numerical linear algebra is used in scientific computing, engineering, and computational physics.
"Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient."
"The algorithms studied here can be used to compute such an approximation."
"An alternative method is to use techniques from calculus to obtain a series expansion of the solution."
"Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics."
"Some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved." Please note that the paragraph provided does not contain twenty distinct ideas or concepts. If you would like more questions or quotes based on this paragraph, please let me know.