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Chapter 6: Q1E (page 331)

In Exercises 1-4, find a least-sqaures solution of \(A{\bf{x}} = {\bf{b}}\) by (a) constructing a normal equations for \({\bf{\hat x}}\) and (b) solving for \({\bf{\hat x}}\).

1. \(A = \left[ {\begin{aligned}{{}{}}{ - {\bf{1}}}&{\bf{2}}\\{\bf{2}}&{ - {\bf{3}}}\\{ - {\bf{1}}}&{\bf{3}}\end{aligned}} \right]\), \({\bf{b}} = \left[ {\begin{aligned}{{}{}}{\bf{4}}\\{\bf{1}}\\{\bf{2}}\end{aligned}} \right]\)

Short Answer

Expert verified

(a) \(\left[ {\begin{aligned}{{}{}}{ - 4}\\{11}\end{aligned}} \right]\)

(b)\(\left[ {\begin{aligned}{{}{}}3\\2\end{aligned}} \right]\)

Step by step solution

01

(a) Step 1: Find the products \({A^T}A\) and \({A^T}{\bf{b}}\)

Find the product \({A^T}A\).

\(\begin{aligned}{}{A^T}A &= \left[ {\begin{aligned}{{}{}}{ - 1}&2&{ - 1}\\2&{ - 3}&3\end{aligned}} \right]\left[ {\begin{aligned}{{}{}}{ - 1}&2\\2&{ - 3}\\{ - 1}&3\end{aligned}} \right]\\ &= \left[ {\begin{aligned}{{}{}}6&{ - 11}\\{ - 11}&{22}\end{aligned}} \right]\end{aligned}\)

Find the product \({A^T}{\bf{b}}\).

\(\begin{aligned}{}{A^T}{\bf{b}} &= \left[ {\begin{aligned}{{}{}}{ - 1}&2&{ - 1}\\2&{ - 3}&3\end{aligned}} \right]\left[ {\begin{aligned}{{}{}}4\\1\\2\end{aligned}} \right]\\ &= \left[ {\begin{aligned}{{}{}}{ - 4}\\{11}\end{aligned}} \right]\end{aligned}\)

02

Find the solution by constructing the normal equations

The normal equations can be written as:

\(\begin{aligned}{}\left( {{A^T}A} \right){\bf{x}} = {A^T}{\bf{b}}\\\left[ {\begin{aligned}{{}{}}6&{ - 11}\\{ - 11}&{22}\end{aligned}} \right]\left[ {\begin{aligned}{{}{}}{{x_1}}\\{{x_2}}\end{aligned}} \right] = \left[ {\begin{aligned}{{}{}}{ - 4}\\{11}\end{aligned}} \right]\end{aligned}\)

03

(b) Step 3: Find the component \({\bf{\hat x}}\)

The component \({\bf{\hat x}}\) can be calculated as:

\(\begin{aligned}{}{\bf{\hat x}} &= {\left( {{A^T}A} \right)^{ - 1}}\left( {{A^T}{\bf{b}}} \right)\\ &= {\left[ {\begin{aligned}{{}{}}6&{ - 11}\\{ - 11}&{22}\end{aligned}} \right]^{ - 1}}\left[ {\begin{aligned}{{}{}}{ - 4}\\{11}\end{aligned}} \right]\\ &= \frac{1}{{11}}\left[ {\begin{aligned}{{}{}}{22}&{11}\\{11}&6\end{aligned}} \right]\left[ {\begin{aligned}{{}{}}{ - 4}\\{11}\end{aligned}} \right]\\ &= \frac{1}{{11}}\left[ {\begin{aligned}{{}{}}{33}\\{22}\end{aligned}} \right]\\ &= \left[ {\begin{aligned}{{}{}}3\\2\end{aligned}} \right]\end{aligned}\)

The \({\bf{\hat x}}\) component is \(\left[ {\begin{aligned}{{}{}}3\\2\end{aligned}} \right]\).

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Most popular questions from this chapter

To measure the take-off performance of an airplane, the horizontal position of the plane was measured every second, from \(t = 0\) to \(t = 12\). The positions (in feet) were: 0, 8.8, 29.9, 62.0, 104.7, 159.1, 222.0, 294.5, 380.4, 471.1, 571.7, 686.8, 809.2.

a. Find the least-squares cubic curve \(y = {\beta _0} + {\beta _1}t + {\beta _2}{t^2} + {\beta _3}{t^3}\) for these data.

b. Use the result of part (a) to estimate the velocity of the plane when \(t = 4.5\) seconds.

In Exercises 1-4, find a least-sqaures solution of \(A{\bf{x}} = {\bf{b}}\) by (a) constructing a normal equations for \({\bf{\hat x}}\) and (b) solving for \({\bf{\hat x}}\).

4. \(A = \left( {\begin{aligned}{{}{}}{\bf{1}}&{\bf{3}}\\{\bf{1}}&{ - {\bf{1}}}\\{\bf{1}}&{\bf{1}}\end{aligned}} \right)\), \({\bf{b}} = \left( {\begin{aligned}{{}{}}{\bf{5}}\\{\bf{1}}\\{\bf{0}}\end{aligned}} \right)\)

Suppose \(A = QR\), where \(R\) is an invertible matrix. Showthat \(A\) and \(Q\) have the same column space.

Let \({{\bf{u}}_1},......,{{\bf{u}}_p}\) be an orthogonal basis for a subspace \(W\) of \({\mathbb{R}^n}\), and let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be defined by \(T\left( x \right) = {\rm{pro}}{{\rm{j}}_W}x\). Show that \(T\) is a linear transformation.

Suppose radioactive substance A and B have decay constants of \(.02\) and \(.07\), respectively. If a mixture of these two substances at a time \(t = 0\) contains \({M_A}\) grams of \(A\) and \({M_B}\) grams of \(B\), then a model for the total amount of mixture present at time \(t\) is

\(y = {M_A}{e^{ - .02t}} + {M_B}{e^{ - .07t}}\) (6)

Suppose the initial amounts \({M_A}\) and are unknown, but a scientist is able to measure the total amounts present at several times and records the following points \(\left( {{t_i},{y_i}} \right):\left( {10,21.34} \right),\left( {11,20.68} \right),\left( {12,20.05} \right),\left( {14,18.87} \right)\) and \(\left( {15,18.30} \right)\).

a.Describe a linear model that can be used to estimate \({M_A}\) and \({M_B}\).

b. Find the least-squares curved based on (6).
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