Advanced Linear Algebra
Advanced linear algebra mostly covers all the advanced topics related to linear algebra such as Linear function, Linear transformation, Eigenvectors, and Eigenvalues, etc.
Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors are fundamental concepts in linear algebra. It offers deep insights into the properties of linear transformations. An eigenvector of a square matrix is a non-zero vector that, when the matrix multiplies it, results in a scalar multiple of itself. This scalar is known as the eigenvalue associated with the eigenvector. They are essential in various applications, including stability analysis, quantum mechanics, and the study of dynamical systems.
Consider a transformation that changes the direction or length of vectors, except for some special vectors that only get stretched or shrunk. These special vectors are eigenvectors, and the factor by which they are stretched or shrunk is the eigenvalue.
Example: For the matrix A = [2, 0, 0, 3], the vector v = 1,0 is an eigenvector because Av = 2v, and 2 is the eigenvalue.
Singular Value Decomposition
Singular Value Decomposition (SVD) is a powerful mathematical technique used in signal processing, statistics, and machine learning. It decomposes a matrix into three other matrices, where one represents the rotation, another the scaling, and the third the final rotation. It’s essential for identifying the intrinsic geometric structure of data.
Linear Algebra
Linear Algebra is the branch of mathematics that focuses on the study of vectors, vector spaces, and linear transformations. It deals with linear equations, linear functions, and their representations through matrices and determinants. It has a wide range of application in Physics and Mathematics. It is the basic concept for machine learning and data science. We have explained the Linear Algebra, types of Linear Algebra.
Let’s learn about Linear Algebra, like linear function, including its branches, formulas, and examples.