Linear Algebra is a branch of mathematics that deals with linear equations, matrices, vectors, and their properties. It is used to solve problems in various fields such as physics, engineering, and computer science.
Linear Algebra has many applications in fields such as data analysis, computer graphics, optimization, and machine learning. It is also used in solving systems of equations, finding eigenvalues and eigenvectors, and performing transformations in geometry.
A matrix is a rectangular array of numbers or symbols. In Linear Algebra, matrices are used to represent linear transformations and solve systems of linear equations. They are also used to perform operations such as addition, subtraction, multiplication, and inversion.
A vector is a mathematical object that has magnitude and direction. In Linear Algebra, vectors are used to represent quantities that have both magnitude and direction, such as velocity and force. They can also be used to represent points in space and perform operations such as addition, subtraction, and scalar multiplication.
Eigenvalues and eigenvectors are important concepts in Linear Algebra. Eigenvalues are the values that satisfy the equation Av = λv, where A is a square matrix, v is a vector, and λ is a scalar. Eigenvectors are the corresponding vectors that are scaled by the eigenvalue λ. They are used in many applications such as solving systems of differential equations and diagonalizing matrices.