Patlam81 said:Let says I have a matrix A with m rows and n columns, with m<n, from which I compute the null space. If the rank of A is smaller than m, then the null space of the transpose of A also exists. Is there any relation between the null space of a matrix and the null space of the transposed matrix? Can I find null(transpose(A)) from null(A) or I have to compute the nullspace again?
The nullspace of a transposed matrix is the set of all vectors that, when multiplied by the transposed matrix, result in a zero vector. In other words, it is the set of all solutions to the equation ATx = 0, where AT represents the transposed matrix and x is a vector of appropriate dimensions.
The nullspaces of a matrix and its transpose are closely related. In fact, they are the same set of vectors. This means that if a vector x is in the nullspace of a matrix A, then it is also in the nullspace of its transpose, AT. This is because the nullspace is determined by the columns of the matrix, and the columns of A and AT are the same.
Yes, it is possible for the nullspace of a transposed matrix to be empty. This happens when the columns of the matrix are linearly independent, meaning that there is no combination of them that can result in a zero vector. In this case, there are no solutions to the equation ATx = 0, and therefore the nullspace is empty.
The nullspace of a transposed matrix can be used to find solutions to linear systems of equations. This is because the nullspace represents all of the possible solutions to a homogeneous system of equations, where the right-hand side is set to zero. It can also be used in data compression and image processing, as it can help identify redundant or irrelevant information in a dataset.
To calculate the nullspace of a transposed matrix, we can use row reduction techniques to put the matrix in reduced row echelon form (RREF). The columns corresponding to the leading ones in the RREF will form a basis for the nullspace. Alternatively, we can use the fact that the nullspaces of a matrix and its transpose are the same to calculate the nullspace of the matrix directly, without transposing it.