To determine whether the sets $S_1$ and $S_2$ span the same subspace of $\mathbb{R}^3$, we need to analyze whether the vectors in $S_1$ and $S_2$ span the same space. Specifically, we want to check if the two sets of vectors are linearly independent and if they span the same subspace of $\mathbb{R}^3$.

Step 1: Check Linear Independence of $S_1$ and $S_2$

First, we'll check the linear independence of the vectors in $S_1$ and $S_2$. If the vectors in each set are linearly independent, they will span a subspace of $\mathbb{R}^3$. If any set has linearly dependent vectors, we will find the rank (dimension) of the subspace it spans.

Set $S_1$:

The vectors in $S_1$ are: $v_1 = (0, 0, 1), \quad v_2 = (0, 1, 1), \quad v_3 = (8, 1, 1)$

To check if these vectors are linearly independent, we form a matrix with these vectors as columns and row reduce it:

0 & 0 & 8 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$$ Perform row reduction on $$A_1$$. 1. Swap rows 1 and 3 to move the non-zero element in row 3 to the top: $$\begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 8 \end{pmatrix}$$ 2. The matrix is now in row echelon form, and all rows contain a pivot, indicating that the vectors in $$S_1$$ are linearly independent. Therefore, the vectors in $$S_1$$ span a 3-dimensional subspace of $$\mathbb{R}^3$$. #### Set $$S_2$$: The vectors in $$S_2$$ are: $$v_4 = (1, 1, 1), \quad v_5 = (1, 1, 5), \quad v_6 = (5, 1, 1)$$ We perform similar row reduction for the matrix formed by $$S_2$$'s vectors: $$A_2 = \begin{pmatrix} 1 & 1 & 5 \\ 1 & 1 & 1 \\ 1 & 5 & 1 \end{pmatrix}$$ 1. Subtract row 1 from rows 2 and 3: $$\begin{pmatrix} 1 & 1 & 5 \\ 0 & 0 & -4 \\ 0 & 4 & -4 \end{pmatrix}$$ 2. Now, divide row 2 by $$-4$$ and subtract it from row 3: $$\begin{pmatrix} 1 & 1 & 5 \\ 0 & 0 & 1 \\ 0 & 4 & 0 \end{pmatrix}$$ 3. Subtract 5 times row 2 from row 1: $$\begin{pmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 4 & 0 \end{pmatrix}$$ 4. Finally, subtract 4 times row 3 from row 2: $$\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ This is the identity matrix, which means the vectors in $$S_2$$ are also linearly independent. Therefore, the vectors in $$S_2$$ span a 3-dimensional subspace of $$\mathbb{R}^3$$. ### Step 2: Compare Spanned Subspaces Since both sets $$S_1$$ and $$S_2$$ contain three linearly independent vectors, they each span a 3-dimensional subspace of $$\mathbb{R}^3$$. In fact, in $$\mathbb{R}^3$$, any set of three linearly independent vectors spans the entire space. Therefore, both $$S_1$$ and $$S_2$$ span the entire space $$\mathbb{R}^3$$. ### Conclusion: Yes, the sets $$S_1$$ and $$S_2$$ span the same subspace of $$\mathbb{R}^3$$—they both span all of $$\mathbb{R}^3$$. Would you like any further details or clarifications? Here are some follow-up questions: 1. How can we verify if a set of vectors spans a subspace other than $$\mathbb{R}^3$$? 2. What is the concept of linear dependence, and how does it relate to the span of a set of vectors? 3. How does the row-reduction process help in determining the rank of a matrix? 4. How do we determine if a set of vectors spans a subspace in higher-dimensional spaces? 5. What are the implications of having a set of linearly dependent vectors in $$\mathbb{R}^3$$? **Tip:** When checking if vectors span $$\mathbb{R}^3$$, remember that you can always reduce a matrix formed by those vectors to see if it has a full rank (rank = 3 for $$\mathbb{R}^3$$).