# Math Help - Linear system

1. ## Linear system

Let A be m x n, (m<n) and of full rank (ie rank(A) = m). Then, the linear system Ax = b has infinite many solutions. Acutally, the general solution depends on n - m arbitrary parameters. How do you find the unique soultion iwth the minumum 2-norm?

2. Originally Posted by tinng
Let A be m x n, (m<n) and of full rank (ie rank(A) = m). Then, the linear system Ax = b has infinite many solutions. Actually, the general solution depends on n - m arbitrary parameters. How do you find the unique solution with the minimum 2-norm?
The general solution will be of the form $\{x_0+v:v\in V\}$, where $x_0$ is a particular solution and V is an (n-m)-dimensional subspace. One way to minimise the 2-norm of $x_0+v$ is to construct an orthonormal basis $\{e_1,\ldots,e_{n-m}\}$ of V. The closest point to $x_0$ in V is then $v_0 = \sum_{j=1}^{n-m}\langle x_0,e_j\rangle e_j$, and the solution with smallest 2-norm is $x_0-v_0$.