8. For the given linear transformations, determine all vectors \(\vec{v}\) such that \(T(\vec{v}) = \vec{0}\) 1. \(T: \mathbb{R}^2 \to \mathbb{R}^2\) defined by \(T(v_1, v_2) = (v_1 - 2v_2, -3v_1 + 8v_2)\). 2. \(T: \mathbb{R}^2 \to \mathbb{R}^2\) defined by \(T(v_1, v_2) = (v_1 + 2v_2, 3v_1 + 6v_2)\).
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$T: R^{2} \rightarrow R^{2}$ is defined by $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{c} x_{1}+7 x_{2} \\ 3 x_{1}-4 x_{2} \end{array}\right] $$ and $B=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$ and $B^{\prime}=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\},$ where $$ \mathbf{u}_{1}=\left[\begin{array}{l} 2 \\ 2 \end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{r} 4 \\ -1 \end{array}\right] ; \quad \mathbf{v}_{1}=\left[\begin{array}{l} 18 \\ 8 \end{array}\right], \quad \mathbf{v}_{2}=\left[\begin{array}{r} 10 \\ 5 \end{array}\right] $$
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