Question: Let \(V\) be the space of all lower-triangular \(2 \times 2\) matrices. Consider the linear transformation

where \(R=\left[\begin{array}{ll}3 & 0 \\ 2 & 1\end{array}\right]\).

1. Find the matrix \(A\) of \(T\) with respect to the basis
   \[\mathcal{B}=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right\}\]
2. Find a basis for the kernel of \(T\).
3. Find a basis for the image of T.

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