## Linear Algebra and Its Applications, Exercise 3.3.26

Exercise 3.3.26. A middle-aged man is stretched on a rack with various forces. Given the measurements of length $L = 5, 6, 7$ (in feet) at forces $F= 1, 2, 4$ (in tons), and assuming that Hooke’s Law $a + bF$ applies, use least squares to find the man’s length $a$ when no force is applied.

Answer: This corresponds to a system of the form $Ax = b$ as follows: $\begin{bmatrix} 1&1 \\ 1&2 \\ 1&4 \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$

To find the least squares solution we multiply both sides by $A^T$ to create a system of the form $A^TA\bar{x} = A^Tb$. We have $A^TA = \begin{bmatrix} 1&1&1 \\ 1&2&4 \end{bmatrix} \begin{bmatrix} 1&1 \\ 1&2 \\ 1&4 \end{bmatrix} = \begin{bmatrix} 3&7 \\ 7&21 \end{bmatrix}$

and $A^Tb = \begin{bmatrix} 1&1&1 \\ 1&2&4 \end{bmatrix} \begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix} = \begin{bmatrix} 18 \\ 45 \end{bmatrix}$

so that the new system is $\begin{bmatrix} 3&7 \\ 7&21 \end{bmatrix} \begin{bmatrix} \bar{a} \\ \bar{b} \end{bmatrix} = \begin{bmatrix} 18 \\ 45 \end{bmatrix}$

We can multiply the first equation by $\frac{7}{3}$ and subtract it from the second equation to form the system $\begin{bmatrix} 3&7 \\ 0&\frac{14}{3} \end{bmatrix} \begin{bmatrix} \bar{a} \\ \bar{b} \end{bmatrix} = \begin{bmatrix} 18 \\ 3 \end{bmatrix}$

From the second equation we have $\bar{b} = 3 \cdot \frac{3}{14} = \frac{9}{14}$ and can substitute into the first equation to get $3\bar{a} + 7 \cdot \frac{9}{14} = 18$ or $\bar{a} = \frac{1}{3} (18 - \frac{9}{2}) = \frac{1}{3} (\frac{27}{2}) = \frac{9}{2}$

The man’s length when not stretched is thus $\bar{a} = \frac{9}{2}$ or 4.5 feet.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fifth Edition and the accompanying free online course, and Dr Strang’s other books .

This entry was posted in linear algebra and tagged . Bookmark the permalink.