a. The original constitutive operator mutations in the lac operon were all base changes in $O_{1} .$ Why do you think mutations in $O_{2}$ or $O_{3}$ were not isolated in these screens? b. Explain how a mutagen that causes small insertions could produce an $O^{c}$ mutation. c. Would the $O^{c}$ mutation described above in part b be sensitive to $\mathrm{lacl}^{s} ?$ Why or why not?

Use the formula $h=\frac{d}{\cot \beta-\cot \alpha}$ to solve the following problem.

A camera operator sits in a stationary crane, 40 feet above ground. He films a scene below of an automobile approaching. At the beginning of the scene, the angle of depression to the driver is $23^{\circ}$. After a direct approach towards the crane, the scene ends with the camera at an angle of $47^{\circ}$. How far did the vehicle travel?

Show that each of the following matrices is orthogonal and find the rotation and/or reflection it produces as an operator acting on vectors. If a rotation, find the axis and angle; if a reflection, find the reflecting plane and the rotation, if any, about the normal to that plane.

$$\frac{1}{2}\left(\begin{array}{ccc} 1 & \sqrt{2} & -1 \\ \sqrt{2} & 0 & \sqrt{2} \\ 1 & -\sqrt{2} & -1 \end{array}\right)$$

Define a linear operator $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ by

$$T\left(\left[\begin{array}{l} x \\ y \end{array}\right]\right)=\left[\begin{array}{r} x-2 y \\ -2 x+4 y \end{array}\right]$$

Determine whether the vector v is in N(T).

$$\mathbf{v}=\left[\begin{array}{l} \frac{1}{2} \\ \frac{1}{4} \end{array}\right]$$

Let $T: V \rightarrow V$ be the linear operator defined by T(v)=Av. a. Show that T is an isomorphism. b. Find $A^{-1}$. c. Show directly that $T^{-1}(\mathbf{w})=A^{-1} \mathbf{w}$ for all $\mathbf{w} \in V$.

$$T\left(\left[\begin{array}{l} x \\ y \end{array}\right]\right)=\left[\begin{array}{rr} -2 & 3 \\ -1 & -1 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]$$

Find whether the linear operator $T: M_{22} \rightarrow M_{22}$ is one-to-one. If so, determine

$$T^{-1}\left(\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\right)$$

$$T\left(\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\right)=\left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right]$$

Let $L$ be a second order linear differential operator. Show that the solution $y=\varphi(x)$ of the problem $L[y]=f(x),\alpha_1y(0)+\alpha_2y'(0)=a,\beta_1y(1)+\beta_2y'(1)=b$ can be written as $y=u+v$, where $u=\varphi1(x)$ and $v=\varphi2(x)$ are solutions of the problems $L[u]=0,\alpha_1u(0)+\alpha_2u'(0)=a,\beta_1u(1)+\beta_2u'(1)=b$ and $L[v]=f(x),\alpha_1v(0)+\alpha_2v'(0)=0,\beta_1v(1)+\beta_2v'(1)=0$,respectively.

Define a linear operator $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ by

$$T\left(\left[\begin{array}{l} x \\ y \\ z \end{array}\right]\right)=\left[\begin{array}{rr} x+ & 2 z \\ 2 x+y+3 z \\ x-y+3 z \end{array}\right]$$

Determine whether the vector v is in R(T).

$$\mathbf{v}=\left[\begin{array}{l} 1 \\ 3 \\ 0 \end{array}\right]$$

Find whether the linear operator $T: M_{22} \rightarrow M_{22}$ is one-to-one. If so, determine

$$T^{-1}\left(\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\right)$$

$$T\left(\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\right)=\left[\begin{array}{ll}a & c \\ b & d\end{array}\right]$$

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Let $T: R^3 \rightarrow R^3$ be the linear operator defined by

$$T\left(x_1, x_2, x_3\right)=\left(x_2-x_3, x_1+x_2, x_3-x_2\right)$$

Determine the matrix for $T$ with respect to the basis $B=\left\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\right\}$, where

$$\mathbf{v}_1=(1,0,1), \quad \mathbf{v}_2=(0,1,1), \quad \mathbf{v}_3=(1,1,0)$$

Find whether the linear operator $T: M_{22} \rightarrow M_{22}$ is one-to-one. If so, determine

$$T^{-1}\left(\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\right)$$

$$T\left(\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\right)=\left[\begin{array}{ll}a & 0 \\ 0 & d\end{array}\right]$$

A linear operator T and bases $B_{1}$ and $B_{2}$ are given. a. Find $[T]_{B_{1}}$ and $[T]_{B_{2}}$. b. Verify that the action on v of the linear operator T is the same when using the matrix representation of T relative to the bases $B_{1}$ and $B_{2}$.

$$T\left(\left[\begin{array}{l} x \\ y \\ z \end{array}\right]\right)=\left[\begin{array}{l} x \\ 0 \\ z \end{array}\right]$$

$B_{1}=\left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right\}$

$$B_{2}=\left\{\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right]\right\}$$

$$\mathbf{v}=\left[\begin{array}{r} 1 \\ 2 \\ -1 \end{array}\right]$$

Consider a physical system whose Hamiltonian $H$ and an operator $A$ are given, in a threedimensional space, by the matrices

$$\begin{align*} H=\hbar \omega\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array}\right), \quad A=a\left(\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right) . \end{align*}$$