| # Set of solutions to the least-squares problem via QR decomposition The set of solutions to the least-squares problem where , and are given, can be expressed in terms of the full QR decomposition of : where is upper triangular and invertible, , is a permutation matrix, and is and orthogonal. Precisely we have , with a matrix whose columns span the nullspace of : Proof: Since and are orthogonal, we have, with , : Exploiting the fact that leaves Euclidean norms invariant, we express the original least-squares problem in the equivalent form: Once the above is solved, and is found, we recover the original variable with . Now let us decompose and in a manner consistent with the block structure of : , , with two -vectors. Then which leads to the following expression for the objective function: The optimal choice for the variables is to make the first term zero, which is achievable with where is free, and describes the ambiguity in the solution. The optimal residual is . We are essentially done: with , we can write that is: , with