【題組】

接下來前段是背景知識介紹,之後才是提問)In solving the linear equation Ax=b for a igiven

, instead of using the elemnentary row operations (i.e. the Gauss eliminations) to

manipulate the equation, we may also apply the QR factorization to the equation to get QRx = b,

which implies further QTQRx = QTb. Since QTQ = In, it gives Rx = QTb. Thus, according to the

result of(a), when all columns of A are linearly independent, the square matrix R is nonsingular and

so the solution x =

b is obtained.

It seems that we may summarize the above argument as the following statement:

Given

, where all columns of A are assumed linearly independent, thensolution to the equation Ax = o can always be computed from x =

, where Q and R arematrices obtained from the @R factorization of A.

 However, the simple example

shows that the summary is incorrect because, accordingto the summary, the solution is x=

 and obviously it does notsatisfy the original equation.(b) (5%) What is the error (or are the errors) in the argument right before the summary to makeit incorrect?