## GATE Data Science & AI Linear Algebra Syllabus

Linear Algebra: Vector space, subspaces, linear dependence and independence of vectors, matrices, projection matrix, orthogonal matrix, idempotent matrix, partition matrix and their properties, quadratic forms, systems of linear equations and solutions; Gaussian elimination, eigenvalues and eigenvectors, determinant, rank, nullity, projections, LU decomposition, singular value decomposition.

Here’s a brief explanation of each:

**Vector Space**: A vector space is a set of vectors along with operations of addition and scalar multiplication that satisfy certain properties such as closure under addition and scalar multiplication, associativity, commutativity, existence of additive identity and inverses, and distributive properties.**Subspaces**: Subspaces are subsets of a vector space that are themselves vector spaces. They must contain the zero vector, be closed under addition and scalar multiplication, and satisfy other properties of vector spaces.**Linear Dependence and Independence of Vectors**: A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. If a vector in the set can be expressed as a linear combination of the others, the set is linearly dependent.**Matrices**: Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. They are used to represent linear transformations, systems of linear equations, and other mathematical objects.**Projection Matrix**: A projection matrix is a square matrix that, when multiplied by a vector, projects that vector onto a subspace.**Orthogonal Matrix**: An orthogonal matrix is a square matrix whose rows and columns are orthogonal unit vectors. It preserves lengths and angles, and its inverse is equal to its transpose.**Idempotent Matrix**: An idempotent matrix is a square matrix that, when multiplied by itself, yields itself.**Partition Matrix**: A partition matrix is a matrix that has been divided into sections or partitions.**Quadratic Forms**: Quadratic forms are homogeneous polynomials of degree two in a set of variables. They are often represented by symmetric matrices and are used in optimization and physics.**Systems of Linear Equations and Solutions**: Systems of linear equations involve multiple linear equations with the same variables. The goal is to find values of the variables that satisfy all the equations simultaneously.**Gaussian Elimination**: Gaussian elimination is a method for solving systems of linear equations by transforming the augmented matrix into row-echelon form through a sequence of row operations.**Eigenvalues and Eigenvectors**: Eigenvalues and eigenvectors are properties of square matrices. An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, yields a scalar multiple of itself (the eigenvalue).**Determinant**: The determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the matrix. It is used in solving systems of linear equations, computing inverses, and determining the behavior of linear transformations.**Rank and Nullity**: The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. The nullity is the dimension of the null space, which consists of all vectors that the matrix maps to zero.**LU Decomposition**: LU decomposition factors a matrix as the product of a lower triangular matrix (L) and an upper triangular matrix (U). It is used to simplify the process of solving systems of linear equations and computing determinants.**Singular Value Decomposition (SVD)**: SVD is a factorization of a matrix into the product of three matrices, one of which is diagonal and contains the singular values of the original matrix. SVD has applications in data compression, noise reduction, and solving linear least squares problems.

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