Linear independence is a fundamental concept in linear algebra. A set of vectors is linearly independent if there isn't a linear combination of them that equals the vector zero, if the combination exists then they are considered to be linearly dependent. Another way to determine linear independence is with a matrix determinant, if the determinant is 0 than they are linearly dependent and if it isn´t then they are linearly independent.

The Role of Linear Independence in Linear Algebra

Linear independence plays a crucial role in linear algebra, as it is used in many different areas of the field. For example, in the study of linear equations, linear independence is used to determine the unique solution of a set of equations. In addition, linear independence is used in the study of vector spaces, where it is used to determine the dimension of a vector space.

In the study of matrices, linear independence is used to determine the invertibility of a matrix. A matrix is invertible if and only if its columns are linearly independent. In addition, linear independence is used in the study of eigenvalues and eigenvectors, where it is used to determine the diagonalizability of a matrix.

Conclusion

Linear independence can be used to solve linear equations and is also related to matrix operations, eigenvalues and eigenvectors. It is important to understand these things because they are a fundamental part of Machine Learning, A.I, some graph algorithms (matroids in the Tutte Polynomial) and several other theories in physics that affect our everyday life.