1. **State the problem:** Determine if the vector $\mathbf{v} = (2, 4, 6, 7, 8)$ is in the subspace of $\mathbb{R}^5$ spanned by the vectors $\mathbf{u}_1 = (1, 2, 0, 3, 0)$, $\mathbf{u}_2 = (0, 0, 1, 4, 0)$, and $\mathbf{u}_3 = (0, 0, 0, 0, 1)$. 2. **Formula and approach:** A vector $\mathbf{v}$ is in the span of $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$ if there exist scalars $a, b, c$ such that: $$ a\mathbf{u}_1 + b\mathbf{u}_2 + c\mathbf{u}_3 = \mathbf{v} $$ This means: $$ a(1,2,0,3,0) + b(0,0,1,4,0) + c(0,0,0,0,1) = (2,4,6,7,8) $$ 3. **Set up the system of equations:** From each coordinate: - $a = 2$ - $2a = 4$ - $b = 6$ - $3a + 4b = 7$ - $c = 8$ 4. **Solve step-by-step:** - From the first equation, $a = 2$. - Check second equation: $2a = 2 \times 2 = 4$, matches the right side. - From third equation, $b = 6$. - Check fourth equation: $3a + 4b = 3 \times 2 + 4 \times 6 = 6 + 24 = 30$, but right side is 7, so this does not match. - From fifth equation, $c = 8$. 5. **Conclusion:** Since the fourth coordinate equation $3a + 4b = 7$ is not satisfied (we get 30 instead of 7), the vector $\mathbf{v}$ is **not** in the subspace spanned by $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$.