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Chapter 2: Q2.9-26E (page 93)
Suppose columns 1, 3, 5, and 6 of a matrix A are linearly independent (but are not necessarily pivot columns) and the rank of A is 4. Explain why the four columns mentioned must be a basis for the column space of A.
The dimension of the column space of matrix A is 4.
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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.)
Suppose A is invertible. Explain why \({A^T}A\) is also invertible. Then show that \({A^{ - {\bf{1}}}} = {\left( {{A^T}A} \right)^{ - {\bf{1}}}}{A^T}\).
In exercise 5 and 6, compute the product \(AB\) in two ways: (a) by the definition, where \(A{b_{\bf{1}}}\) and \(A{b_{\bf{2}}}\) are computed separately, and (b) by the row-column rule for computing \(AB\).
\(A = \left( {\begin{aligned}{*{20}{c}}{\bf{4}}&{ - {\bf{2}}}\\{ - {\bf{3}}}&{\bf{0}}\\{\bf{3}}&{\bf{5}}\end{aligned}} \right)\), \(B = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{3}}\\{\bf{2}}&{ - {\bf{1}}}\end{aligned}} \right)\)
Exercises 1-4 display sets in \({\mathbb{R}^2}\). Assume the sets include the bounding lines. In each case, give a specific reason why the set H is not a subspace of \({\mathbb{R}^2}\). (For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.)
2.
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 \({\mathop{\rm d}\nolimits} = \left[ {\begin{array}{*{20}{c}}{40}\\{60}\\{80}\end{array}} \right]\).
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