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[Step-by-Step] Suppose that T: V \rightarrow V is a linear operator on a three dimensional vector space V with basis \mathcalB. Suppose that for every v ∈

Question: Suppose that \(T: V \rightarrow V\) is a linear operator on a three dimensional vector space \(V\) with basis \(\mathcal{B}\). Suppose that for every \(\mathbf{v} \in V\) we have

\[[T(\mathbf{v})]_{\mathcal{B}}=\left(\begin{array}{c} x_{1}-x_{2}+x_{3} \\ x_{1}+x_{2} \\ x_{1}-x_{2} \end{array}\right) \quad \text { where } \quad[\mathbf{v}]_{\mathcal{B}}=\left(\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right)\]

Find the matrix \([T]_{\mathcal{B}, \mathcal{B}}\) for \(T\) with respect to the basis \(\mathcal{B}\). Is \(\operatorname{ker}(T)=\{0\} ?\) Is \(R(T)=V ?\)

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