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Chapter 6: Q23E (page 331)

Suppose \(A\) is \(m \times n\) with linearly independent columns and \(b\) is in \({\mathbb{R}^m}\). Use the normal equations to produce a formula for \(\hat b\), the projection of b onto \({\rm{Col}}\,A\).

Short Answer

Expert verified

The formula is given by, \(\hat b = A{\left( {{A^T}A} \right)^{ - 1}}{A^T}b\).

Step by step solution

01

Least-square solution

It is given that,\(A\) is a \(m \times n\) matrix with linearly independent columns andb is in \({\mathbb{R}^m}\). So, the least-square solution \(\hat x\)is given by \(\hat x = {\left( {{A^T}A} \right)^{ - 1}}{A^T}b\).

02

Formula for \(\hat b\)

Given that\(b\)is an orthogonal projection onto\({\rm{Col}}\,A\), then\(\hat b = A\hat x\).

Substitute\(\hat x = {\left( {{A^T}A} \right)^{ - 1}}{A^T}b\)in the above equation to get:

\(\begin{aligned}{}\hat b &= A\hat x\\\hat b &= A{\left( {{A^T}A} \right)^{ - 1}}{A^T}b\end{aligned}\)

Hence, the required formula is \(\hat b = A{\left( {{A^T}A} \right)^{ - 1}}{A^T}b\).

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Most popular questions from this chapter

In exercises 1-6, determine which sets of vectors are orthogonal.

\(\left[ {\begin{array}{*{20}{c}}5\\{ - 4}\\0\\3\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}{ - 4}\\1\\{ - 3}\\8\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}3\\3\\5\\{ - 1}\end{array}} \right]\)

Find an orthogonal basis for the column space of each matrix in Exercises 9-12.

12. \(\left( {\begin{aligned}{{}{}}1&3&5\\{ - 1}&{ - 3}&1\\0&2&3\\1&5&2\\1&5&8\end{aligned}} \right)\)

Find an orthogonal basis for the column space of each matrix in Exercises 9-12.

11. \(\left( {\begin{aligned}{{}{}}1&2&5\\{ - 1}&1&{ - 4}\\{ - 1}&4&{ - 3}\\1&{ - 4}&7\\1&2&1\end{aligned}} \right)\)

Compute the quantities in Exercises 1-8 using the vectors

\({\mathop{\rm u}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{ - 1}\\2\end{aligned}} \right),{\rm{ }}{\mathop{\rm v}\nolimits} = \left( {\begin{aligned}{*{20}{c}}4\\6\end{aligned}} \right),{\rm{ }}{\mathop{\rm w}\nolimits} = \left( {\begin{aligned}{*{20}{c}}3\\{ - 1}\\{ - 5}\end{aligned}} \right),{\rm{ }}{\mathop{\rm x}\nolimits} = \left( {\begin{aligned}{*{20}{c}}6\\{ - 2}\\3\end{aligned}} \right)\)

8. \(\left\| {\mathop{\rm x}\nolimits} \right\|\)

A healthy child鈥檚 systolic blood pressure (in millimetres of mercury) and weight (in pounds) are approximately related by the equation

\({\beta _0} + {\beta _1}\ln w = p\)

Use the following experimental data to estimate the systolic blood pressure of healthy child weighing 100 pounds.

\(\begin{array} w&\\ & {44}&{61}&{81}&{113}&{131} \\ \hline {\ln w}&\\vline & {3.78}&{4.11}&{4.39}&{4.73}&{4.88} \\ \hline p&\\vline & {91}&{98}&{103}&{110}&{112} \end{array}\)
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