Algebraic applications of the algorithm

How to obtain the inverse of a matrix

In this lesson it is explained how the orthogonalization algorithm can be used to obtain the inverse of a matrix. At the same time, the determinant of the matrix is obtained. The pproblem of updating the inverse and the determinant of a matrix after changing a row is deal with. It is demonstrated that a single step allows this updating without the need of repeating the whole process.

Rank of a matrix

In this leson we describe how the orthogonalization algorithm can be used to obtain the rank of a matrix. The method is very simple, because we determine if a row or column vector of the matrix belongs to the subspace of the previous uones and even we obtain the coefficients for the linear combination. The method is much simpler than the one based on obtaining the minors, a large collection of determinants that must be non null to get the rank.

How to determine if a vector belongs to a linear subspace

In this lesson we explain how to use the orthogonal algorithm to know if one or more vectors belong to a linear subspace. The method is very simple and fast. It is based on the concept of orthogonality, that is, we use the point of view of orthogonality to solve the problem.

Intersection of two subspaces

In this lesson we use the orthogonalization algorithm to obtain the intersection of two subspaces. We use the point of view of orthoganilty to realize thet the intersection is the orthogonal subspace to the dual of the first subspace in the second subspace or the orthogonal subspace to the dual of the second subspace in the first subspace