Linear Algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves the use of matrices, vectors, and other mathematical structures to solve problems related to linear systems.
For a system of linear equations to have a unique solution, the number of equations must be equal to the number of variables. Additionally, the system must be consistent, meaning that there is at least one solution that satisfies all of the equations, and the equations must be linearly independent, meaning that no equation can be derived from a combination of the others.
A consistent system of linear equations has at least one solution that satisfies all of the equations, while an inconsistent system has no solution that satisfies all of the equations. In other words, a consistent system has a unique solution or an infinite number of solutions, while an inconsistent system has no solutions.
A matrix is invertible if it is a square matrix (number of rows equals number of columns) and its determinant is non-zero. This means that the matrix must have an inverse, which is a matrix that, when multiplied by the original matrix, results in the identity matrix (a matrix with 1s on the main diagonal and 0s everywhere else).
Linear independence is a condition that ensures a matrix is invertible. If a matrix has linearly independent rows or columns, it means that the matrix is full rank and has a non-zero determinant, which are both necessary for a matrix to be invertible. In other words, if a matrix is not linearly independent, it cannot be inverted.
