**Problem** **16.1:** (4.3 #17. *Introduction* *to* *Linear* *Algebra:* Strang) Write down three equations for the line *b* = *C* + *Dt* to go through *b* = 7 at *t* = −1, *b* = 7 at *t* = 1, and *b* = 21 at *t* = 2. Find the least squares solution **xˆ **= (*C*, *D*) and draw the closest line.

**Problem** **16.2:** (4.3 #18.) Find the projection **p** = *A***xˆ **in the previous problem. This gives the three heights of the closest line. Show that the error vector is **e** = (2, −6, 4). Why is *P***e** = **0**?

**Problem** **16.3:** (4.3 #19.) Suppose the measurements at *t* = −1,1,2 are the errors 2, -6, 4 in the previous problem. Compute **xˆ **and the closest line to these new measurements. Explain the answer: **b** = (2, −6, 4) is perpendicular to so the projection is **p** = **0**.

**Problem** **16.4:** (4.3 #20.) Suppose the measurements at *t* = −1,1,2 are **b** = (5, 13, 17). Compute **xˆ **and the closest line and **e**. The error is **e** = **0** because this **b** is .

**Problem** **16.5:** (4.3 #21.) Which of the four subspaces contains the error vector **e**? Which contains **p**? Which contains **xˆ**? What is the nullspace of *A*?

**Problem** **16.6:** (4.3 #22.) Find the best line *C* + *Dt* to fit *b* = 4, 2, −1,0,0 at times *t* = −2, −1, 0, 1, 2.

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