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

Study the chart showing the different functions that political parties fulfill. Choose two functions, and write a paragraph each about how modern political parties in your area fulfill those purposes.

Optional Exam 1 Definition Retry: D1: State the definition of a linear combination (of matrices). D2: State the definition of a linear transformation. Optional Exam 2 Definition Retry: D3: Complete the following definition: If \(S=\left\{v_{1},v_{2},\ldots,v_{k}\right\}\) is a set of vectors in a vector space \(V\), then \(S\) is linearly independent if: D4: Complete the following definition: If \(S=\left\{v_{1},v_{2},\ldots,v_{k}\right\}\) is a set of vectors in a vector space \(V\), then \(S\) is a basis for \(V\) if:

Which of the following lists the steps in filling a Supreme Court vacancy? 

How did the Supreme Court gain the power of judicial review?

Find a basis for the solution space of the following system, and give its dimension. \[\begin{align}w+x+3y-2z&=0\\2w+3x-3y-5z&=0\end{align}\]  

Given the following matrix \(A\) and its rref, find bases for \(\text{row}\left(A\right), \text{col}\left(A\right),\) and \(\text{null}\left(A\right)\). Also, state the rank and nullity of \(A\). \[A=\begin{bmatrix}1&3&2&7&1&-1&3\\3&9&1&17&-3&-12&-3\\-1&-3&0&-5&1&-21&1\\-1&-3&5&5&4&0&7\end{bmatrix}\] \[\text{rref}\left(A\right)=\begin{bmatrix}1&3&0&5&0&1&1\\0&0&1&2&0&-3&0\\0&0&0&0&1&4&2\\0&0&0&0&0&0&0\end{bmatrix}\]

In 2015, the Supreme Court ruled that ___________.

The quadriceps femoris muscle group _________ the hip and __________the knee.

Which type of opinion represents the winning side of a Supreme Couth case, when not all of the justices agree?