1. **Problem statement:** Find scalars $a$ and $b$ such that $$a\mathbf{u} + b\mathbf{v} = (1, -4, 9, 18)$$ where $$\mathbf{u} = (1, -1, 3, 5)$$ and $$\mathbf{v} = (2, 1, 0, -3).$$ 2. **Set up the equation component-wise:** $$a(1) + b(2) = 1$$ $$a(-1) + b(1) = -4$$ $$a(3) + b(0) = 9$$ $$a(5) + b(-3) = 18$$ 3. **Write the system of linear equations:** $$\begin{cases} a + 2b = 1 \\ - a + b = -4 \\ 3a = 9 \\ 5a - 3b = 18 \end{cases}$$ 4. **Solve for $a$ from the third equation:** $$3a = 9 \implies a = \frac{9}{3} = 3$$ 5. **Substitute $a=3$ into the first equation:** $$3 + 2b = 1 \implies 2b = 1 - 3 = -2 \implies b = -1$$ 6. **Check the second equation with $a=3$, $b=-1$:** $$-3 + (-1) = -4$$ which is true. 7. **Check the fourth equation:** $$5(3) - 3(-1) = 15 + 3 = 18$$ which is true. 8. **Conclusion:** The scalars are $$a = 3$$ and $$b = -1$$. **Final answer:** $$\boxed{a=3, b=-1}$$