Given 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 \(A\). \[A=\begin{bmatrix}2&1&-1&0&3&8&3\\3&1&-3&3&-8&19&7\\-1&1&5&-2&8&-11&-2\\-1&1&5&1&1&20&6\end{bmatrix}\] \[\text{rref}\left(A\right)=\begin{bmatrix}1&0&-2&0&1&5&1\\0&1&3&0&1&-2&1\\0&0&0&1&-4&2&1\\0&0&0&0&0&0&0\end{bmatrix}\]

Let \(V=\mathbb{R}^{2}\) with the following operations \[\be…

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