Exam 2 Definitions 1.) [5 points each] Complete both of the following definitions. You ONLY have to write out what correctly finishes the definition, not the part that is given. a.) If \(S=\left\{v_{1},\ldots ,v_{k}\right\}\) is a set of vectors in a vector space \(V\), then \(S\) is linearly independent if: b.) If \(S=\left\{v_{1},\ldots ,v_{k}\right\}\) is a set of vectors in a vector space \(V\), then \(S\) is a basis for \(V\) if: The use of math tags around this note triggers MathJax to display the LaTex written on this page as MathML objects.

Which constitutional amendment allows citizens to feel safe from illegal searches and seizures?

How has the relationship between mass media and government changed in the past few decades?  Explain whether changes in technology have given the media more or less power to influence public policy and cite specific examples. (2 points)

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 \(A\). \[A=\begin{bmatrix}1&3&2&7&1&-1&3\\3&9&1&17&-3&-12&-3\\-1&-3&0&-5&1&-21&1\\-1&-3&5&5&4&0&7\end{bmatrix}\] \[\text{rref}\left(A\right)=\begin{bmatrix}1&3&0&5&0&1&1\\0&0&1&2&0&-3&0\\0&0&0&0&1&4&2\\0&0&0&0&0&0&0\end{bmatrix}\]