1. **State the problem:** Find the linear transformation $T: V_3(\mathbb{R}) \to V_3(\mathbb{R})$ determined by the matrix $$ \begin{bmatrix} 1 & 2 & 1 \\ 0 & 1 & 1 \\ -1 & 3 & 4 \end{bmatrix} $$ with respect to the standard basis $\{e_1, e_2, e_3\}$. 2. **Recall the standard basis:** The standard basis vectors for $V_3(\mathbb{R})$ are $$ e_1 = \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}, \quad e_2 = \begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix}, \quad e_3 = \begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}. $$ 3. **Definition of the linear transformation:** The matrix $A = \begin{bmatrix}1 & 2 & 1 \\ 0 & 1 & 1 \\ -1 & 3 & 4\end{bmatrix}$ defines $T$ by $$ T(x) = A x $$ for any vector $x \in V_3(\mathbb{R})$. 4. **Find $T(e_1)$:** Multiply $A$ by $e_1$: $$ T(e_1) = A e_1 = \begin{bmatrix}1 & 2 & 1 \\ 0 & 1 & 1 \\ -1 & 3 & 4\end{bmatrix} \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ -1\end{bmatrix}. $$ 5. **Find $T(e_2)$:** Multiply $A$ by $e_2$: $$ T(e_2) = A e_2 = \begin{bmatrix}1 & 2 & 1 \\ 0 & 1 & 1 \\ -1 & 3 & 4\end{bmatrix} \begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix} = \begin{bmatrix}2 \\ 1 \\ 3\end{bmatrix}. $$ 6. **Find $T(e_3)$:** Multiply $A$ by $e_3$: $$ T(e_3) = A e_3 = \begin{bmatrix}1 & 2 & 1 \\ 0 & 1 & 1 \\ -1 & 3 & 4\end{bmatrix} \begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix} = \begin{bmatrix}1 \\ 1 \\ 4\end{bmatrix}. $$ 7. **Interpretation:** The linear transformation $T$ sends the standard basis vectors as follows: $$ T(e_1) = \begin{bmatrix}1 \\ 0 \\ -1\end{bmatrix}, \quad T(e_2) = \begin{bmatrix}2 \\ 1 \\ 3\end{bmatrix}, \quad T(e_3) = \begin{bmatrix}1 \\ 1 \\ 4\end{bmatrix}. $$ This completely determines $T$ on $V_3(\mathbb{R})$ because any vector $v = x e_1 + y e_2 + z e_3$ is mapped by linearity: $$ T(v) = x T(e_1) + y T(e_2) + z T(e_3). $$ **Final answer:** The linear transformation $T$ determined by the given matrix sends the standard basis vectors as above.