1. **State the problem:** We need to check if the transformation $$T(x_1,x_2,x_3) = (x_1 - x_2, x_2 - x_3, x_1)$$ is a linear transformation. 2. **Recall the definition of linear transformation:** A transformation $$T: \mathbb{R}^n \to \mathbb{R}^m$$ is linear if for all vectors $$\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$$ and scalars $$c$$, the following two properties hold: - Additivity: $$T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$$ - Homogeneity (scalar multiplication): $$T(c\mathbf{u}) = cT(\mathbf{u})$$ 3. **Check additivity:** Let $$\mathbf{u} = (u_1,u_2,u_3)$$ and $$\mathbf{v} = (v_1,v_2,v_3)$$. Calculate $$T(\mathbf{u} + \mathbf{v})$$: $$ T(u_1+v_1, u_2+v_2, u_3+v_3) = ((u_1+v_1) - (u_2+v_2), (u_2+v_2) - (u_3+v_3), u_1+v_1) $$ Simplify: $$ = (u_1 - u_2 + v_1 - v_2, u_2 - u_3 + v_2 - v_3, u_1 + v_1) $$ Calculate $$T(\mathbf{u}) + T(\mathbf{v})$$: $$ T(u_1,u_2,u_3) + T(v_1,v_2,v_3) = (u_1 - u_2, u_2 - u_3, u_1) + (v_1 - v_2, v_2 - v_3, v_1) $$ $$ = (u_1 - u_2 + v_1 - v_2, u_2 - u_3 + v_2 - v_3, u_1 + v_1) $$ Since both expressions are equal, additivity holds. 4. **Check homogeneity:** Let $$c$$ be a scalar. Calculate $$T(c\mathbf{u})$$: $$ T(cu_1, cu_2, cu_3) = (cu_1 - cu_2, cu_2 - cu_3, cu_1) = c(u_1 - u_2, u_2 - u_3, u_1) $$ Calculate $$cT(\mathbf{u})$$: $$ c(u_1 - u_2, u_2 - u_3, u_1) = c(u_1 - u_2, u_2 - u_3, u_1) $$ Since both expressions are equal, homogeneity holds. 5. **Conclusion:** Both additivity and homogeneity hold, so $$T$$ is a linear transformation. **Final answer:** $$T(x_1,x_2,x_3) = (x_1 - x_2, x_2 - x_3, x_1)$$ is a linear transformation.