Give the structure of the polyurethane formed by the reaction of toluene diisocyanate with bisphenol A.

Prove that the given function is a linear transformation. $T: C[0,1] \rightarrow \mathbf{R}^{2}$ with

$$T(f)=\left[\begin{array}{l} f(0) \\ f(1) \end{array}\right]$$

Find an example that meets the given specifications. A linear transformation $T: \mathrm{R}^{2} \rightarrow \mathrm{R}^{2}$ such that

$$T\left(\left[\begin{array}{l} 0 \\ 1 \end{array}\right]\right)=\left[\begin{array}{l} 1 \\ 4 \\ 5 \end{array}\right]$$

Prove that the given function is a linear transformation. $T: V \rightarrow W$ with $T(\mathbf{v})=\mathbf{0}_{W}$ for all v.

Simplify each expression. 5a - 2 - 3a + 7

Find a formula for the quadratic form with the given matrix A.

$$A=\left[\begin{array}{llll} 5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 0 \\ 1 & 0 & 0 & 3 \\ 0 & 0 & 3 & 4 \end{array}\right]$$

The lipid(s) used as the basis of vitamin D, sex hormones, and bile salts is/are (a) triglycerides, (b) cholesterol, (c) phospholipids, (d) prostaglandin.

In adults, over 90% of all cancers are either adenomas (adenocarcinomas) or carcinomas. In fact, cancers of the skin, lung, colon, breast, and prostate are all in these categories. Which one of the four basic tissue types gives rise to most cancers? Why do you think this is so?

Describe the transformation from the graph of $f(x)=\sqrt x$ to the graph of $h(x)=2\sqrt{x-1}+4$.

The graph of $h$ is a translation ______ unit(s) _______, then ______by a factor of ______, and last a translation ______ unit(s) ______ of the graph of $f$.

• up

• down

• left

• right

• vertical stretch

• vertical shrink

• horizontal stretch

• horizontal shrink

• 1/2

• 1

• 4

• 2

Find the determinant of the matrix.

$$\left[\begin{array}{ll} 2 & 1 \\ 3 & 4 \end{array}\right]$$

Find the set of solutions for the given linear system. Note that some systems have only one equation.

$$\begin{array}{r} 3 x_{1}-2 x_{2}+x_{3}=4 \\ -6 x_{3}=-12 \end{array}$$

Construct the integers $\mathbb{Z}$ from the natural numbers $\mathbb{N}$ in a method similar to that used in the Exercise by defining an appropriate equivalence relation on $\mathbb{N} \times \mathbb{N}$.

The quantitative data set under consideration has roughly a bell-shaped distribution. Apply the empirical rule to solve each exercise. The data set has mean 15 and standard deviation 2. Approximately what percentage of the observations lie between 13 and 17?

Write the following systems using vectors and matrices.

$$\left\{\begin{array}{l} 2 x+5 y-z=2 \\ -x+2 y=-1 \\ x+4 z=0 \end{array}\right.$$