1. **Problem statement:** Given vectors $V_1 = (3, 2, 0)$, $V_2 = (1, 3, 0)$, $V_3 = (2, 5, 0)$, and $V_4 = (6, 1, 0)$ in the set $V = \{(a,b,0) \mid a,b \in \mathbb{R}\}$, find three linear combinations of these vectors. 2. **Recall the definition of a linear combination:** A linear combination of vectors $V_1, V_2, V_3, V_4$ is any vector of the form $$c_1 V_1 + c_2 V_2 + c_3 V_3 + c_4 V_4 = (c_1 3 + c_2 1 + c_3 2 + c_4 6, c_1 2 + c_2 3 + c_3 5 + c_4 1, 0)$$ where $c_1, c_2, c_3, c_4$ are real numbers. 3. **Find three example linear combinations by choosing coefficients:** - Combination 1: Let $c_1=1, c_2=0, c_3=0, c_4=0$ $$1 \cdot V_1 = (3, 2, 0)$$ - Combination 2: Let $c_1=0, c_2=1, c_3=1, c_4=0$ $$0 \cdot V_1 + 1 \cdot V_2 + 1 \cdot V_3 + 0 \cdot V_4 = (1+2, 3+5, 0) = (3, 8, 0)$$ - Combination 3: Let $c_1=2, c_2=-1, c_3=0, c_4=1$ $$2 \cdot V_1 - 1 \cdot V_2 + 0 \cdot V_3 + 1 \cdot V_4 = (2\times3 -1\times1 + 0 + 6, 2\times2 -1\times3 + 0 + 1, 0) = (6 -1 + 6, 4 -3 + 1, 0) = (11, 2, 0)$$ 4. **Summary:** The three linear combinations are: $$ (3, 2, 0), (3, 8, 0), (11, 2, 0) $$ These are vectors in $V$ formed by linear combinations of $V_1, V_2, V_3, V_4$.