## Linear Algebra and Its Applications, Exercise 3.3.18

Exercise 3.3.18. Suppose we have the following measurements of $y$: $y = 3, t = 1, z = 1$ $y = 5, t = 2, z = 1$ $y = 6, t = 0, z = 3$ $y = 0, t = 0, z = 0$

and want to fit a plane of the form $y = C + Dt + Ez$.

a) Write a system of 4 equations in 3 unknowns representing the problem. (The system may not have a solution.)

b) Write a system of 3 equations in 3 unknowns  representing the least squares solution to the problem.

Answer: a) The original problem can be expressed as the following system $Ax = b$ of 4 equations in 3 unknowns: $\begin{bmatrix} 1&1&1 \\ 1&2&1 \\ 1&0&3 \\ 1&0&0 \end{bmatrix} \begin{bmatrix} C \\ D \\ E \end{bmatrix} = \begin{bmatrix} 3 \\ 5 \\ 6 \\ 0 \end{bmatrix}$

From the fourth equation we have $C=0$ and then substituting into the third equation we have $E=2$. However we then get differing values of $D$ depending on whether we substitute into the second equation or the first. The system has no solution $x$ as written.

b) The least square solutions amounts to solving the system $A^TA\bar{x} = A^Tb$ that minimizes the error vector $b - A\bar{x}$. We have $A^TA = \begin{bmatrix} 1&1&1&1 \\ 1&2&0&0 \\ 1&1&3&0 \end{bmatrix} \begin{bmatrix} 1&1&1 \\ 1&2&1 \\ 1&0&3 \\ 1&0&0 \end{bmatrix}$ $= \begin{bmatrix} 4&3&5 \\ 3&5&3 \\ 5&3&11 \end{bmatrix}$

and $A^Tb = \begin{bmatrix} 1&1&1&1 \\ 1&2&0&0 \\ 1&1&3&0 \end{bmatrix} \begin{bmatrix} 3 \\ 5 \\ 6 \\ 0 \end{bmatrix} = \begin{bmatrix} 14 \\ 13 \\ 26 \end{bmatrix}$

so that the resulting system of 3 equations in 3 unknowns $A^TA\bar{x} = A^Tb$ is $\begin{bmatrix} 4&3&5 \\ 3&5&3 \\ 5&3&11 \end{bmatrix} \begin{bmatrix} \bar{C} \\ \bar{D} \\ \bar{E} \end{bmatrix} = \begin{bmatrix} 14 \\ 13 \\ 26 \end{bmatrix}$

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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