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, then solution to the equation Ax = o can always be computed from x = 
, where Q and R are matrices obtained from the @R factorization of A.
However, the simple example
shows that the summary is incorrect because, according to the summary, the solution is x= 
and obviously it does not satisfy the original equation. (b) (5%) What is the error (or are the errors) in the argument right before the summary to make it incorrect?
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where the Fourier transform of f1(t) is given in the following figure (a).
where the Fourier transform of f2(t) is given in the following figure (b).
mean? Give your answer necessary explanations.
and denote X := [x1,... ,xn].Let X = QR be the QR factorization of matrix X. Is there any relationship between matrices T, Q and R? If yes, write an equation to describe such a relationship. If no, give a brief explanation for it.
≠ 0 and u 0. Find the solution of x1 and x2 for the initial conditions
.
.Find the range of k such that the solution of x1 and x2 converges to zero for any initial condition.
.Find the range of k such that the solution of x1 and x2exhibits oscillatory behavior for any nonzero initial condition.
, calculate the steady-state response of y(t) = 2x1(t)-x2(t).