1. **Problem Statement:** Given vectors $v_1$ and $v_2$, list seven vectors in the span of $\{v_1, v_2\}$, show the weights used to generate each vector, and list their entries. Also, provide a geometric description of the span. 2. **Span Definition:** The span of vectors $v_1$ and $v_2$ is the set of all linear combinations of these vectors: $$\text{Span}\{v_1, v_2\} = \{a v_1 + b v_2 \mid a,b \in \mathbb{R}\}$$ where $a$ and $b$ are scalar weights. 3. **Example Vectors:** Suppose $v_1 = \begin{bmatrix}x_1 \\ y_1 \\ z_1\end{bmatrix}$ and $v_2 = \begin{bmatrix}x_2 \\ y_2 \\ z_2\end{bmatrix}$. Then any vector in the span is: $$v = a v_1 + b v_2 = \begin{bmatrix}a x_1 + b x_2 \\ a y_1 + b y_2 \\ a z_1 + b z_2\end{bmatrix}$$ 4. **Seven Vectors in Span:** Choose seven pairs $(a,b)$ and compute $v$: - For $(a,b) = (1,0)$: $v = 1 v_1 + 0 v_2 = v_1 = \begin{bmatrix}x_1 \\ y_1 \\ z_1\end{bmatrix}$ - For $(a,b) = (0,1)$: $v = 0 v_1 + 1 v_2 = v_2 = \begin{bmatrix}x_2 \\ y_2 \\ z_2\end{bmatrix}$ - For $(a,b) = (1,1)$: $v = v_1 + v_2 = \begin{bmatrix}x_1 + x_2 \\ y_1 + y_2 \\ z_1 + z_2\end{bmatrix}$ - For $(a,b) = (2,-1)$: $v = 2 v_1 - v_2 = \begin{bmatrix}2 x_1 - x_2 \\ 2 y_1 - y_2 \\ 2 z_1 - z_2\end{bmatrix}$ - For $(a,b) = (-1,2)$: $v = -v_1 + 2 v_2 = \begin{bmatrix}-x_1 + 2 x_2 \\ -y_1 + 2 y_2 \\ -z_1 + 2 z_2\end{bmatrix}$ - For $(a,b) = (0.5,0.5)$: $v = 0.5 v_1 + 0.5 v_2 = \begin{bmatrix}0.5 x_1 + 0.5 x_2 \\ 0.5 y_1 + 0.5 y_2 \\ 0.5 z_1 + 0.5 z_2\end{bmatrix}$ - For $(a,b) = (-1,-1)$: $v = -v_1 - v_2 = \begin{bmatrix}-x_1 - x_2 \\ -y_1 - y_2 \\ -z_1 - z_2\end{bmatrix}$ 5. **Geometric Description:** The span of two vectors $v_1$ and $v_2$ in $\mathbb{R}^3$ is a plane passing through the origin that contains both $v_1$ and $v_2$. Every vector in this plane can be expressed as a linear combination of $v_1$ and $v_2$. This completes the listing and explanation of vectors in the span and the geometric interpretation.