(b) (8%) Obviously, matrix T' relates to the two bases E and F. What conclusions about vectors in basis E and/or in basis F can be drawn if the matrix Tis known to be diagonal?

題組內容

4.(14%)下面的問題共有二個子題,只要簡短扼要地回答提問即可,不須寫出答案背後的推導。
Let V be a vector space with dim(V) = n and an ordered basis E := [x₁, ..,x_n].Let F := [y₁,... , Y_n] be the ordered orthonormal basis generated from basis E by applying the Gran- Schmidt Orthogonalization Process. For any, let [v]_Eand [v]_F denote the coordinate vectors of v with respect to bases E and F, respectively. Let T denote the transition matrix from basis E to basis F. Let L : , ie. L is a linear operator mapping V into itself, and suppose that
Let's denote the matrix representation of L with respect to basis E by A.

(a) (6%) Please continue the argument to derive the result rank(A) = rank(R). (接下來前段是背景知識介紹,之後才是提問)Insthq given, instead of using the elementary 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 QT QRx = 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 = 6 can always be computed from x =, where Q and R are matrices obtained from the QR factorization of A. However, the simple example shows that the summary is incorrect because, according to the summary, the solution is x ==0 and obviously it does not satisfy the original equation.

題組內容

(b) (8%) Obviously, matrix T' relates to the two bases E and F. What conclusions about vectors in basis E and/or in basis F can be drawn if the matrix Tis known to be diagonal?

相關申論題

相關試卷