In what basic way can all people influence public policy?

Which constitutional amendment guarantees freedom of the press?

The use of math tags around this note triggers MathJax to display the LaTex written on this page as MathML objects. Complete each of the following definitions:  D1.) [4 points] If \(A\) is a square matrix, then \(A\) is diagonalizable if: D2.) [6 points] If \(V\) is a real vector space, then \(\left\langle ,\right\rangle\) is an inner product on \(V\) if: (there are four properties) 1.) [10 pts] Let \(A=\begin{bmatrix}3&-2\\-1&2\end{bmatrix}\). Find the eigenvalues of \(A\), and then find a basis for ONE of the eigenspaces. (Your choice. Do NOT find bases for both, I will only grade the first complete one that you have worked.) 2.) [10 pts] Let \(B=\begin{bmatrix}6&0&0\\1&-3&0\\2&5&-1\end{bmatrix}\). Find the eigenvalues of \(B\), and then find a basis for any ONE of the eigenspaces. (Your choice. Do NOT find bases for all, I will only grade the first complete one that you have worked.) 3.) [6 pts each] Let \(\overrightarrow{u}=\begin{bmatrix}2\\1\\-2\end{bmatrix}\) and \(\overrightarrow{v}=\begin{bmatrix}-1\\5\\-4\end{bmatrix}\) be vectors in \(\mathbb{R}^{3}\), with weighted Euclidean inner product corresponding to the weights \(w_{1}=1\), \(w_{2}=2\), and \(w_{3}=4\). Find each of the following. a.) The angle \(\theta\) between \(\overrightarrow{u}\) and \(\overrightarrow{v}\), to the nearest tenth of a degree. b.) \(\text{proj}_{\overrightarrow{v}}\overrightarrow{u}\). 4.) [6 pts] Using \(\overrightarrow{a}=\begin{bmatrix}2\\1\\-2\end{bmatrix}\) and \(\overrightarrow{b}=\begin{bmatrix}-1\\5\\-4\end{bmatrix}\), but now with inner product corresponding to the matrix \(A=\begin{bmatrix}1&0&-1\\0&-1&2\\1&-1&1\end{bmatrix}\), find the inner product \(\left\langle\overrightarrow{a},\overrightarrow{b}\right\rangle\). 5.) [6 pts each] Let \(f=1+x\) and \(g=1+3x^{2}\).  a.) Treating \(g\) as an element of \(P_{2}\), find \(\left\Vert g\right\Vert\), using the evaluation inner product, with test points \(x_{0}=0\), \(x_{1}=1\), and \(x_{2}=-1\). b.) Treating \(f\) and \(g\) as elements of \(C\left[0,1\right]\), find \(\left\langle f,g\right\rangle\), using the integral inner product on \(C\left[0,1\right]\). 6.) [6 pts] Let \(A=\begin{bmatrix}1&-5&2\\-1&5&1\end{bmatrix}\) and \(B=\begin{bmatrix}-1&0&3\\2&4&3\end{bmatrix}\) be elements of \(M_{2\,3}\) with its standard inner product. Find \(\left\langle A,B\right\rangle\). 7.) [10 pts] Let \(W\) be the subspace of \(\mathbb{R}^{4}\) with the following basis. \[\left\{\begin{bmatrix}-1\\0\\0\\2\end{bmatrix},\begin{bmatrix}1\\1\\2\\0\end{bmatrix},\begin{bmatrix}1\\2\\-1\\1\end{bmatrix}\right\}\] Use the Gram-Schmidt process to convert this to an orthogonal basis for \(W\). (Hint: If you clear fractions while going through the orthogonalization process, as shown in our examples, it makes things easier.) You are to use the standard inner product (i.e. the dot product) here. 8.) [10 pts] Let \(W\) be the subspace of \(\mathbb{R}^{4}\) with the following orthogonal basis. \[\left\{\begin{bmatrix}2\\-1\\2\\-1\end{bmatrix},\begin{bmatrix}1\\3\\3\\5\end{bmatrix}\right\}\] Let \(\overrightarrow{v}=\begin{bmatrix}1\\-1\\0\\-4\end{bmatrix}\). Find the projection \(\text{proj}_{W}\overrightarrow{v}\). 9.) Consider the inconsistent linear system \[\begin{align}2x+y&=5\\x-2y&=0\\x-2y&=3\\-x+y&=3\end{align}\] a.) [6 pts] Find the associated normal system. This should be written in the form of a system of linear equations, not in matrix form. b.) [4 pts] Find the least squares solution. c.) [4 pts] Find the least squares error. EXTRA CREDIT. [5 points] If \(W\) is a subspace of \(\mathbb{R}^{n}\) whose basis vectors are placed as columns into the matrix \(A\), what is the matrix \(P_{W}\) that projects vectors into \(W\)? That is, what is the formula, in terms of \(A\), for the matrix \(P_{W}\) that has the property: \[P_{W}\overrightarrow{v} = \text{proj}_{W}\overrightarrow{v}\]  

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 \(P_{\mathcal{B}\to\mathcal{B}^{\prime}}\).