
# 7.2: Review Problems

1.    A door factory can buy supplies in two kinds of packages, $$f$$ and $$g$$. The package $$f$$ contains $$3$$ slabs of wood, $$4$$ fasteners, and $$6$$ brackets. The package $$g$$ contains $$5$$ fasteners, $$3$$ brackets, and $$7$$ slabs of wood.

a)    Give a list of inputs and outputs for the functions $$f$$ and $$g$$.

b)    Give an order to the 3 kinds of supplies and then write $$f$$ and $$g$$ as elements of $$\Re^{3}$$.

c)    Let $$L$$ be the manufacturing process; it takes in supply packages and gives out two products (doors, and door frames) and it is linear in supplies. If $$Lf$$ is $$1$$ door and $$2$$ frames and $$Lg$$ is $$3$$ doors and $$1$$ frame, find a matrix for L.

2.    You are designing a simple keyboard synthesizer with two keys. If you push the first key with intensity $$a$$ then the speaker moves in time as $$a\sin(t)$$. If you push the second key with intensity $$b$$ then the speaker moves in time as $$b\sin(2t)$$. If the keys are pressed simultaneously,

a)    Describe the set of all sounds that come out of your synthesizer. ($$\textit{Hint:}$$ Sounds can be "added".)

b)    Graph the function $$\begin{pmatrix}3\\5\end{pmatrix}\in \Re^{\{1,2\}}$$.

c)    Let $$B=(\sin(t), \sin(2t))$$. Explain why $$\begin{pmatrix}3\\5\end{pmatrix}_{B}$$ is not in $$\Re^{\{1,2\}}$$ but is still a function.

d)    Graph the function $$\begin{pmatrix}3\\5\end{pmatrix}_{B}$$.

3.
a)    Find the matrix for $$\frac{d}{dx}$$ acting on the vector space $$V$$ of polynomials of degree 2 or less in  the ordered basis $$B'=(x^{2},x,1)$$

b)    Use the matrix from part (a) to rewrite the differential equation $$\frac{d}{dx} p(x)=x$$ as a matrix equation. Find all solutions of the matrix equation. Translate them into elements of $$V$$.

c)    Find  the matrix for $$\frac{d}{dx}$$ acting on the vector space $$V$$  in the ordered basis $$(x^{2}+x,x^{2}-x,1)$$.

d)    Use the matrix from part (c) to rewrite the differential equation $$\frac{d}{dx} p(x)=x$$ as a matrix equation. Find all solutions of the matrix equation. Translate them into elements of $$V$$.

e)    Compare and contrast your results from parts (b) and (d).

4.    Find the "matrix'' for $$\frac{d}{dx}$$ acting on the vector space of all power series in the ordered basis $$(1,x,x^{2},x^{3},...)$$. Use this matrix to find all power series solutions to the differential equation $$\frac{d}{dx} f(x)=x$$. $$\textit{Hint:}$$ your "matrix'' may not have finite size.

5.    Find the matrix for $$\frac{d^{2}}{dx^{2}}$$ acting on $$\{ c_{1} \cos(x)+c_{2} \sin(x) |c_{1},c_{2}\in \Re\}$$ in the ordered basis $$(\cos(x),\sin(x))$$.

6.    Find the matrix for $$\frac{d}{dx}$$ acting on $$\{ c_{1} \cosh(x)+c_{2} \sinh(x) |c_{1},c_{2}\in \Re\}$$ in the ordered basis $$(\cosh(x)+\sinh(x), \cosh(x)-\sinh(x))$$. (Recall that the hyperbolic trigonometric functions are defined by $$\cosh(x)=\frac{e^{x}+e^{-x}}{2}, \sinh(x)=\frac{e^{x}-e^{-x}}{2}$$.)

7.    Let $$B=(1,x,x^{2})$$ be an ordered basis for
$$V=\{ a_{0}+a_{1}x+a_{2}x^{2}| a_{0},a_{1},a_{2} \in \Re\}\, ,$$
and let $$B'=(x^{3},x^{2},x,1)$$ be an ordered basis for
$$W=\{ a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3} | a_{0},a_{1},a_{2},a_{3} \in \Re\}\, ,$$
Find the matrix for the operator $${\cal I}:V\to W$$ defined by $${\cal I}p(x)=\int_{1}^{x} p(t)dt$$ relative to these bases.