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# 4.5: Review Problems

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1. When he was young, Captain Conundrum mowed lawns on weekends to help pay his college tuition bills. He charged his customers according to the size of their lawns at a rate of 5 cent per square foot and meticulously kept a record of the areas of their lawns in an ordered list:
$$A=(200,300,50,50,100,100,200,500,1000,100)\, .$$
He also listed the number of times he mowed each lawn in a given year, for the year 1988 that ordered list was
$$f=(20,1,2,4,1,5,2,1,10,6)\, .$$

a) Pretend that $$A$$ and $$f$$ are vectors and compute $$A\cdot f$$.

b) What quantity does the dot product $A\dotprod f$ measure?

c) How much did Captain Conundrum earn from mowing lawns in 1988? Write an expression for this amount in terms of the vectors $$A$$ and $$f$$.

d) Suppose Captain Conundrum charged different customers different rates. How could you modify the expression in part c) to compute the Captain's earnings?

2.
(2) Find the angle between the diagonal of the unit square in $$\mathbb{R}^{2}$$ and one of the coordinate axes.

(3) Find the angle between the diagonal of the unit cube in $$\mathbb{R}^{3}$$ and one of the coordinate axes.

(n) Find the angle between the diagonal of the unit (hyper)-cube in $$\mathbb{R}^{n}$$ and one of the coordinate axes.

($$\infty$$ What is the limit as $$n \to \infty$$ of the angle between the diagonal of the unit (hyper)-cube in $$\mathbb{R}^{n}$$ and one of the coordinate axes?

3. Consider the matrix
$$M = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{pmatrix}$$ and the vector $$X = \begin{pmatrix}x\\y\end{pmatrix}$$.

a) Sketch $$X$$ and $$MX$$ in $$\mathbb{R}^{2}$$ for several values of $$X$$ and $$\theta$$.
b) Compute $$\frac{||MX||}{||X||}$$ for arbitrary values of $$X$$ and $$\theta$$.
c) Explain your result for (b) and describe the action of $$M$$ geometrically.

​4. (Lorentzian Strangeness). For this problem, consider $$\mathbb{R}^{n}$$ with the Lorentzian inner product defined in example 46 of section 4.3.

a) Find a non-zero vector in two-dimensional Lorentzian space-time with zero length.

b) Find and sketch the collection of all vectors in two-dimensional Lorentzian space-time with zero length.

c) Find and sketch the collection of all vectors in three-dimensional Lorentzian space-time with zero length.

5. Create a system of equations whose solution set is a 99 dimensional hyperplane in $$\Re^{101}$$.

6. Recall that a plane in $$\Re^{3}$$ can be described by the equation
$$n \cdot \begin{pmatrix}x\\ y\\ z\end{pmatrix}=n\cdot p$$
where the vector $$p$$ labels a given point on the plane and $$n$$ is a vector normal to the plane. Let $$N$$ and $$P$$ be vectors in $$\Re^{101}$$ and
$$X=\begin{pmatrix}x^{1}\\x^{2}\\ \vdots\\ x^{101}\end{pmatrix}.$$
What kind of geometric object does $$N\cdot X= N\cdot P$$ describe?

7. Let
$$u=\begin{pmatrix}1\\1\\1\\ \vdots \\ 1\end{pmatrix} {\rm ~and~} v= \begin{pmatrix}1\\2\\3\\ \vdots\\ \! 101\!\end{pmatrix}$$
Find the projection of $$v$$ onto $$u$$ and the projection of $$u$$ onto $$v$$. ($$\textit{Hint:}$$ Remember that two vectors $$u$$ and $$v$$ define a plane, so first work out how to project one vector onto another in a plane. The picture from Section 14.4 could help.)

8. If the solution set to the equation $$A(x)=b$$ is the set of vectors whose tips lie on the paraboloid $$z=x^{2}+y^{2}$$, then what can you say about the function $$A$$?

9. Find a system of equations whose solution set is
$$\left\{ \begin{pmatrix}1\\1\\2\\0\end{pmatrix} +c_1 \begin{pmatrix}-1\\-1\\0\\1\end{pmatrix} +c_2 \begin{pmatrix}0\\0\\-1\\-3\end{pmatrix} \middle| \,c_1,c_2\in \Re \right\}.$$
Give a general procedure for going from a parametric description of a hyperplane to a system of equations with that hyperplane as a solution set.

10. If $$A$$ is a linear operator and both $$x=v$$ and $$x=cv$$ (for any real number $$c$$) are solutions to $$Ax=b$$, then what can you say about $$b$$?