Let W be a subspace of R n, y any vector in R n, and the orthogonal projection of y onto W. …

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Presentation transcript:

Let W be a subspace of R n, y any vector in R n, and the orthogonal projection of y onto W. …

Then is the closest point in W to y, in the sense that for all v in W distinct from. …

The vector in this theorem is called the best approximation to y by elements of W.

Inconsistent systems arise often in applications …

When a solution of Ax = b is demanded and none exists, the best one can do is to find an x that makes Ax as close as possible to b. …

Let us take Ax as an approximation to b. The smaller the distance between b and Ax, given by, the better the approximation. …

The general least- squares problem is to find an x that makes as small as possible. …

The term least- squares arises from the fact that is the square root of a sum of squares.

If A is m x n and b is in R m, a least-squares solution of Ax = b is an in R n such that

The set of least-squares solutions of Ax = b coincides with the nonempty set of solutions of the normal equations

Find a least-squares solution of the inconsistent system Ax = b for

Find a least-squares solution of Ax = b for

The matrix A T A is invertible iff the columns of A are linearly independent. In this case, the equation Ax = b has only one least-squares solution, and it is given by

Determine the least-squares error in the least-squares solution of Ax = b.

Find a least-squares solution of Ax = b for

Given an m x n matrix A with linearly independent columns, let A = QR be a QR factorization of A. …

Then for each b in R m, the equation Ax = b has a unique least-squares solution, given by

Find the least-squares solution of Ax = b for