Linear Algebra and Its Applications, Exercise 3.3.25

Exercise 3.3.25. Given the measurements $y = 2, 0, -3, -5$ at $t= -1, 0, 1, 2$ from the previous exercise, what would be the coefficient matrix $A$, the unknown vector $x$, and the data vector $b$ if we wish to fit the data using a parabola of the form $y = C + Dt + Et^2$?

Answer: We would need to change the unknown vector $x$ to add the additional unknown parameter $E$ and add a third column to the coefficient matrix $A$ to reflect the values of $t^2$ for the various measurements. The vector $b$ containing the data values would remain the same. The system $Ax = b$ would then be as follows:

$\begin{bmatrix} 1&-1&1 \\ 1&0&0 \\ 1&1&1 \\ 1&2&4 \end{bmatrix} \begin{bmatrix} C \\ D \\ E \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \\ -3 \\ -5 \end{bmatrix}$

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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