To identify a basis for the span of the set $S_2 = \{(-1, 0), (-4, 1), (2, 3)\}$, we need to determine whether the vectors are linearly independent and, if necessary, find a minimal set of linearly independent vectors that span the same space.

Step 1: Write the vectors in matrix form

The vectors can be written as columns in a matrix:

-1 & -4 & 2 \\ 0 & 1 & 3 \end{pmatrix}$$ ### Step 2: Row-reduce the matrix We row-reduce the matrix to check for linear independence: $$\begin{pmatrix} -1 & -4 & 2 \\ 0 & 1 & 3 \end{pmatrix}$$ 1. Divide the first row by -1 to make the leading entry 1: $$\begin{pmatrix} 1 & 4 & -2 \\ 0 & 1 & 3 \end{pmatrix}$$ 2. Subtract 4 times the second row from the first row: $$\begin{pmatrix} 1 & 0 & -14 \\ 0 & 1 & 3 \end{pmatrix}$$ This matrix is now in row echelon form. Each row corresponds to a pivot variable, and there are two pivot columns, meaning that the original set contains two linearly independent vectors. ### Step 3: Identify the basis The pivot columns correspond to the vectors $$(-1, 0)$$ and $$(-4, 1)$$. Therefore, these two vectors form a basis for the span of the set. ### Final Answer: The basis for the span of $$S_2$$ is: $$\{(-1, 0), (-4, 1)\}$$ Would you like further clarification on any part of this process? Here are some related questions to explore further: 1. What is the geometric interpretation of the basis vectors in $$\mathbb{R}^2$$? 2. Can a set with more than 2 vectors in $$\mathbb{R}^2$$ ever be linearly independent? 3. How does the row reduction process help identify linearly independent vectors? 4. What is the dimension of the span of $$S_2$$? 5. How can we generalize this method to higher dimensions? **Tip**: When reducing a matrix, always try to simplify row operations to avoid fractions where possible.