Use the Wronskian to show that the set \(\left\{5x,4x^{2},\cos\left(x\right)\right\}\) is linearly independent in \(C^{\infty}\left(-\infty,\infty\right)\).

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 \[\left(\overrightarrow{a}\oplus\overrightarrow{b}\right)\oplus\overrightarrow{c}=\overrightarrow{a}\oplus\left(\overrightarrow{b}\oplus\overrightarrow{c}\right)\]

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Use the Wronskian to show that the set \(\left\{3x,x^{2},\sin\left(x\right)\right\}\) is linearly independent in \(C^{\infty}\left(-\infty,\infty\right)\).