Let \(A\) be an \(m\times n\) matrix and let \(W=\left\{\overrightarrow{x}\in\mathbb{R}^{n}: A\overrightarrow{x}=\overrightarrow{0}\right\}\). (That is, \(W=\text{null}\left(A\right)\).) Prove that \(W\) is a subspace of \(\mathbb{R}^{n}\).  

How are interest groups and political parties similar?

Requiring a photo ID has been shown to most heavily reduce voting among which groups?

If \(\mathcal{B}=\left\{1+x+x^{2},1+x,1\right\}\), find \(\left(p\right)_{\mathcal{B}}\) for the polynomial \(p=8+3x-5x^{2}\).

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

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Let \(\mathcal{B}=\left\{\begin{bmatrix}2\\-3\end{bmatrix},\begin{bmatrix}3\\-5\end{bmatrix}\right\}\) and \(\mathcal{B}^{\prime}=\left\{\begin{bmatrix}5\\4\end{bmatrix},\begin{bmatrix}1\\1\end{bmatrix}\right\}\). Find the transition matrix \(P_{\mathcal{B}\to\mathcal{B}^{\prime}}\).

  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)