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Let $L: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a linear operator. If \[ L\left((1,2)^{T}\right)=(-2,3)^{T} \] and \[ L\left((1,-1)^{T}\right)=(5,2)^{T} \] fi...

Question

Let $L: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a linear operator. If \[ L\left((1,2)^{T}\right)=(-2,3)^{T} \] and \[ L\left((1,-1)^{T}\right)=(5,2)^{T} \] find the value of $L\left((7,5)^{T}\right)$

Let $L: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a linear operator. If \[ L\left((1,2)^{T}\right)=(-2,3)^{T} \] and \[ L\left((1,-1)^{T}\right)=(5,2)^{T} \] find the value of $L\left((7,5)^{T}\right)$



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If $\mathbf{r}(t)=\mathbf{u}(t) \times \mathbf{v}(t),$ where $\mathbf{u}$ and $\mathbf{v}$ are the vector
functions in Exercise $77,$ find $\mathbf{r}^{\prime}(2) .$


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