## Linear Algebra and Its Applications, Exercise 3.3.22

Exercise 3.3.22. Given measurements of $b = 4, 2, -1, 0, 0$ at $t = -2, -1, 0, 1, 2$ use least squares to find the line of best fit of the form $C + Dt$.

Answer: This corresponds to a system $Ax = b$ as follows: $\begin{bmatrix} 1&-2 \\ 1&-1 \\ 1&0 \\ 1&1 \\ 1&2 \end{bmatrix} \begin{bmatrix} C \\ D \end{bmatrix} = \begin{bmatrix} 4 \\ 2 \\ -1 \\ 0 \\ 0 \end{bmatrix}$

To find the least squares solution we form the system $A^TA\bar{x} = A^Tb$. We have $A^TA = \begin{bmatrix} 1&1&1&1&1 \\ -2&-1&0&1&2 \end{bmatrix} \begin{bmatrix} 1&-2 \\ 1&-1 \\ 1&0 \\ 1&1 \\ 1&2 \end{bmatrix}$ $= \begin{bmatrix} 5&0 \\ 0&10 \end{bmatrix}$

and $A^Tb = \begin{bmatrix} 1&1&1&1&1 \\ -2&-1&0&1&2 \end{bmatrix} \begin{bmatrix} 4 \\ 2 \\ -1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 5 \\ -10 \end{bmatrix}$

The system $A^TA\bar{x} = A^Tb$ is then $\begin{bmatrix} 5&0 \\ 0&10 \end{bmatrix} \begin{bmatrix} \bar{C} \\ \bar{D} \end{bmatrix} = \begin{bmatrix} 5 \\ -10 \end{bmatrix}$

From the second equation we have $\bar{D} = -1$ and from the first equation we have $\bar{C} = 1$. The line of best fit is therefore $1 - t$.

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 .

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