Author: Anonymous

States regulate interstate commerce; the national government, intrastate commerce. 

Let \(V=\mathbb{R}^{2}\) with the following operations \[\begin{align} \left(x_{1},y_{1}\right)\oplus\left(x_{2},y_{2}\right)&=\left(x_{1}+x_{2},2y_{1}+2y_{2}\right)\\k\otimes\left(x,y\right)&=\left(kx,ky\right) \end{align}\] Show that \(V\) is not a vector space by showing that its addition rule is not associative, that is, show that it fails the axiom \[\left(\overrightarrow{a}\oplus\overrightarrow{b}\right)\oplus\overrightarrow{c}=\overrightarrow{a}\oplus\left(\overrightarrow{b}\oplus\overrightarrow{c}\right)\]

Why do citizens have a civic duty to serve on juries when called upon?

What did James Madison think of interest groups, or what he called “factions” as expressed in The Federalist No. 10?

Which group can influence the meaning of the Constitution through judicial review?

In what basic way can all people influence public policy?

Under which circumstance might a witness give up his or her right to freedom from self-incrimination? 

Determine whether \(\left\{ \begin{bmatrix}2\\1 \\1\end{bmatrix}, \begin{bmatrix}-1\\4 \\1\end{bmatrix}, \begin{bmatrix}0\\2 \\-3\end{bmatrix}\right\}\) form a basis for \(\mathbb{R}^3\).

Let \(f\) be the transformation of \(\mathbb{R}^{2}\) given by rotating \(50^{\circ}\) counterclockwise around the origin. Find the standard matrix for \(f\) and then use that to find where the point \(\left(1,4\right)\) gets sent under this rotation. (As \(50^{\circ}\) is not a special angle, you will have to use decimal approximations.)

Use the Wronskian to show that the set \(\left\{x,3x^{2},\cos\left(x\right)\right\}\) is linearly independent in \(C^{\infty}\left(-\infty,\infty\right)\).