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Math Help - Linear system

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    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?
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  2. #2
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    Quote Originally Posted by tinng View Post
    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.
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