91Ó°ÊÓ

24. (QR Factorization) Suppose \[A = QR\], where Qand R are \[n \times n\], Ris invertible and upper triangular, and Q has the property that \[{Q^T}{\bf{Q}} = I\]. Show that for each b in \[{\mathbb{R}^n}\], the equation \[Ax = b\] has a unique solution. What computations with Q and R will produce the solution?

The solution to the steady-state heat flow problem for
the plate in the figure is approximated by the solution to the
equation\(A{\bf{x}} = {\bf{b}}\);where\(b = \left( {5,15,0,10,0,10,20,30} \right)\)and

\(A = \left[ {\begin{array}{*{20}{c}}4&{ - 1}&{ - 1}&{}&{}&{}&{}&{}\\{ - 1}&4&0&{ - 1}&{}&{}&{}&{}\\{ - 1}&0&4&{ - 1}&{ - 1}&{}&{}&{}\\{}&{ - 1}&{ - 1}&4&0&{ - 1}&{}&{}\\{}&{}&{ - 1}&0&4&{ - 1}&{ - 1}&{}\\{}&{}&{}&{ - 1}&{ - 1}&4&0&{ - 1}\\{}&{}&{}&{}&{ - 1}&0&4&{ - 1}\\{}&{}&{}&{}&{}&{ - 1}&{ - 1}&4\end{array}} \right]\)

(Refer to Exercise 33 of Section 1.1.) The missing entries in Aare zeros. The nonzero entries of A lie within a band along the main diagonal. Such band matricesoccur in a variety of applications and often are extremely large (with thousands of rows and columns but relatively narrow bands).

1-6: Solve the equation Ax=b by using the LU factorization given for A. In Exercises 1 and 2, also solve \(Ax = b\) by ordinary row reduction.

4. \(A = \left[ {\begin{array}{*{20}{c}}2&{ - 2}&4\\1&{ - 3}&1\\3&7&5\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}0\\{ - 5}\\7\end{array}} \right]\)

\(A = \left[ {\begin{array}{*{20}{c}}1&0&0\\{\frac{1}{2}}&1&0\\{\frac{3}{2}}&{ - 5}&1\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}2&{ - 2}&4\\0&{ - 2}&{ - 1}\\0&0&{ - 6}\end{array}} \right]\)

Verify the boxed statement preceding Example 1 i.e., Let A and B be square matrices. If \[AB = I\], then Aand B are both invertible, with \[B = {A^{ - {\bf{1}}}}\] and \[A = {B^{ - {\bf{1}}}}\].

The consumption matrix C for the U.S. economy in 1972 has the property that every entryin the matrix \({\left( {I - C} \right)^{ - 1}}\) is nonzero (and positive). What does that say about the effect of raising the demand for the output of just one sector of the economy?

Exercises 1-4 refer to an economy that is divided into three sectors - manufacturing, agriculture, and services. For each unit of output, manufacturing requires .10 unit from other companies in that sector, .30 unit from services. For each unit of output, agriculture uses .20 unit of its own output, .60 unit from manufacturing, and .10 unit from services. For each unit of output, the services sector consumes .10 unit from services, .60 unit from manufacturing, but no agricultural products.

3. Determine the production levels needed to satisfy a final demand of 18 units for manufacturing, with no final demand for the other sectors. (Do not compute an inverse matrix.)

Solve the Leontief production equation for an economy with three sectors, given that

\(C = \left[ {\begin{array}{*{20}{c}}{.2}&{.2}&{.0}\\{.3}&{.1}&{.3}\\{.1}&{.0}&{.2}\end{array}} \right]\)and \({\mathop{\rm d}\nolimits} = \left[ {\begin{array}{*{20}{c}}{40}\\{60}\\{80}\end{array}} \right]\).

Consider the following geometric 2D transformations: D, a dilation (in which x-coordinates and y-coordinates are scaled by the same factor); R, a rotation; and T a translation. Does D commute with R? That is, is \(D\left( {R\left( {\bf{x}} \right)} \right) = R\left( {D\left( {\bf{x}} \right)} \right)\)for all \({\bf{x}}\) in \({\mathbb{R}^{\bf{2}}}\)? Does D commute with T? Does R commute with T?

Show that the transformation in Exercise 7 is equivalent to a rotation about the origin followed by a translation by p. Find p.

What vector in \({\mathbb{R}^{\bf{3}}}\) has homogeneous coordinates \(\left( {\frac{1}{2}, - \frac{1}{4},\frac{1}{8},\frac{1}{{24}}} \right)\)?