On page 293, we derived the least squares solution, *b = X+y, to the over determined system Xb = y..

On page
293, we derived the least squares solution, *b = X+y, to the over determined system
Xb = y by use of the QR decomposition of X. This equation for the least squares
solution can also be shown to be correct in other ways ) For simplicity in
this part, let us assume that X is of full rank; that is, we will write *b as
the solution to the normal equations, *b = (XTX)_1XTy. (This assumption
is not necessary, but the proof without it is much messier. Show that *b = X+y
by showing that (XTX)_1XT = X+.

b) On
page 291 we showed that orthogonality of the residuals characterized a least
»

On page
293, we derived the least squares solution, *b = X+y, to the over determined system
Xb = y by use of the QR decomposition of X. This equation for the least squares
solution can also be shown to be correct in other ways ) For simplicity in
this part, let us assume that X is of full rank; that is, we will write *b as
the solution to the normal equations, *b = (XTX)_1XTy. (This assumption
is not necessary, but the proof without it is much messier. Show that *b = X+y
by showing that (XTX)_1XT = X+.

b) On
page 291 we showed that orthogonality of the residuals characterized a least squares
solution. Show that *b = X+y is a least squares solution by showing that in
this case XT(y _ X*b) = 0.

 

 

»

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