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Mathematics LibreTexts

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.

 

 

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