Let A be an n x n matrix such that det(A) = 0, and let

$$T: R^n→R^n$$

be multiplication by A. (a) What can you say about the range of the matrix operator T? Give an example that illustrates your conclusion. (b) What can you say about the number of vectors that T maps into O?

Let V be the space $C^2$, with the standard inner product. Let T be the linear operator defined by $T \epsilon_{1}=(1,-2), T \epsilon_{2}=(i,-1)$. If $\alpha=\left(x_{1}, x_{2}\right)$ find $T^{*} \alpha$.

Suppose

$$T : V \rightarrow V$$

is a linear operator with the property that

$$T [ T ( \mathbf { v } ) ] = \mathbf { v }$$

for all v in V. (For example, transposition in

$$\mathbf { M } _ { n n }$$

or conjugation in

$$\mathbb { C }$$

.) If

$$\mathbf { v } \neq \mathbf { 0 }$$

in V, show that {v, T(v)} is linearly independent if and only if

$$T ( \mathbf { v } ) \neq \mathbf { v }$$

and

$$T ( \mathbf { v } ) \neq - \mathbf { v }$$

.

Suppose $T: P_2 \rightarrow P_2$ be the linear operator defined by

$$T\left(a_0+a_1 \cdot x+a_2 x^2\right)=a_0+a_1(x-1)+a_2(x-1)^2$$

Determine the matrix for $T$ with respect to the standard basis $B=\left\{1, x, x^2\right\}$ for $P_2$.

Determine the dimensions of the kernel and the image of the linear operator T on the space $\mathbb{R}^{n}$ defined by $T\left(x_{1}, \ldots, x_{n}\right)^{\mathrm{t}}=\left(x_{1}+x_{n}, x_{2}+x_{n-1}, \ldots, x_{n}+x_{1}\right)^{t}$

Rewrite the following expressions using the conditional operator. (Assume that all variables are declared properly.) a.

 if (x >= y)
     z = x - y;
else
     z = y - x;

b.

if (hours >= 40.0)
    wages = 40 * 7.50 + 1.5 * 7.5 * (hours - 40);
else
    wages = hours * 7.50;

c.

 if (score >= 60)
    str = "Pass";
else
    str = "Fail";

An amateur radio operator wishes to build a receiver that can tune a range from $14.0 ~\mathrm{MHz}$ to $15.0 ~\mathrm{MHz}$. A variable capacitor has a minimum capacitance of $92$ pF. (a) What is the required value of the inductance? (b) What is the maximum capacitance used on the variable capacitor?

Describe the inverse of the linear operator. (a) Reflection of $R^{2}$ about the x-axis. (b) Rotation of $R^{2}$ about the origin through an angle of $\pi / 4$. (c) Dilation of $R^{2}$ by a factor of 3. (d) Compression of $R^{2}$ in the y-direction with factor $\frac{1}{2}$.

Let $L$ be the linear operator on $\mathbb{R}^3$ defined by

$$L(\mathbf{x})=\left(\begin{array}{l} x_2-x_1 \\ x_3-x_2 \\ x_3-x_1 \end{array}\right)$$

and let $S=\operatorname{Span}\left((1,0,1)^T\right)$.

(a) Determine the kernel of $L$.

Consider the subspace $\mathcal{W}$ of $\mathcal{M}_{n n}$ consisting of all $n \times n$ symmetric matrices, and let B be a fixed $n \times n$ nonsingular matrix. (a) Prove that if $\mathbf{A} \in \mathcal{W}$, then $\mathbf{B}^{T} \mathbf{A} \mathbf{B} \in \mathcal{W}$. (b) Prove that the linear operator on $\mathcal{W}$ given by $L(\mathbf{A})=\mathbf{B}^{T} \mathbf{A B}$ is an isomorphism.

a. If p(t) is a polynomial and $\lambda$ a scalar, show that

$$(D-\lambda)\left(p(t) e^{\lambda t}\right)=p^{\prime}(t) e^{\lambda t}.$$

b. If p(t) is a polynomial of degree less than m, what is

$$(D-\lambda)^{m}\left(p(t) e^{\lambda t}\right) ?$$

c. Find a basis of the kernel of the linear differential operator $(D-\lambda)^{m}.$ d. If $\lambda_{1}, \ldots , \lambda_{r}$ are distinct scalars and $m_{1}, \ldots , m_{r}$ are positive integers, find a basis of the kernel of the linear differential operator

$$\left(D-\lambda_{1}\right)^{m_{1}} \ldots\left(D-\lambda_{r}\right)^{m_{r}}.$$

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

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

Is $T$ one-to-one? If so, find the matrix of $T^{-1}$ with respect to the basis $B$.

A contractor purchases a bulldozer for $36,500. The bulldozer requires an average expenditure of$11.25 per hour for fuel and maintenance, and the operator is paid $19.50 per hour. Write a linear equation giving the total cost C of operating the bulldozer for t hours. (Include the purchase cost of the bulldozer.)

Indicate true or false for the following statements:

(a) Every element in a list must have the same typ
(b) A list’s size is fixed after it is created.
(c) A list can have duplicate elements.
(d) The elements in a list can be accessed via an index operator.