Let \(\mathcal{B}=\left\{\begin{bmatrix}4\\-1\end{bmatrix},\…
Let \(\mathcal{B}=\left\{\begin{bmatrix}4\\-1\end{bmatrix},\begin{bmatrix}7\\-2\end{bmatrix}\right\}\) and \(\mathcal{B}^{\prime}=\left\{\begin{bmatrix}5\\4\end{bmatrix},\begin{bmatrix}1\\1\end{bmatrix}\right\}\). Find the transition matrix \(P_{\mathcal{B}\to\mathcal{B}^{\prime}}\).
Read DetailsGiven the following matrix \(A\) and its rref, find bases fo…
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 \(A\). \[A=\begin{bmatrix}2&1&-1&0&3&8&3\\3&1&-3&3&-8&19&7\\-1&1&5&-2&8&-11&-2\\-1&1&5&1&1&20&6\end{bmatrix}\] \[\text{rref}\left(A\right)=\begin{bmatrix}1&0&-2&0&1&5&1\\0&1&3&0&1&-2&1\\0&0&0&1&-4&2&1\\0&0&0&0&0&0&0\end{bmatrix}\]
Read DetailsLet \(f\) be the transformation of \(\mathbb{R}^{2}\) given…
Let \(f\) be the transformation of \(\mathbb{R}^{2}\) given by rotating \(70^{\circ}\) counterclockwise around the origin. Find the standard matrix for \(f\) and then use that to find where the point \(\left(4,4\right)\) gets sent under this rotation. (As \(70^{\circ}\) is not a special angle, you will have to use decimal approximations.)
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