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Find a formula for the least-squares solution of $A \mathbf{x}=\mathbf{b}$ when the columns of $A$ are orthonormal.

$\hat{\mathbf{x}}=A^{T} \mathbf{b}$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 5

Least-Squares Problems

Vectors

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So we know that if the columns of a out or focal now, then we can compute the projection onto the span of the columns of a off the given back Toby Also noting that a let's say za and by N matrix So the production of B is given by, well, the projection onto each column of a which we denote a I we confusions given by be Doc Aye, aye. Normalized by a I don't a I And once we know what the projection off bees, then it immediate to compute that the least square solution X hat just given by these coefficients and green. So be that a one normalized by a one, not a one all the way down to be taught a n divided by a n dot es en But now if in addition, the columns of a R or for normal, that means that hey, I got a I is just one. So all these denominators in green there just once and so we know that if the columns of a r r for normal than the least square solution except is just given by the given vector of me dot the columns of a

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