 Consider the set in R2
:
S2 = {(−1, 0),(−4, 1),(2, 3)}
Identify a basis for the span of this set.


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:
\[
A = \begin{pmatrix}
-1 & -4 & 2 \\
0  & 1  & 3
\end{pmatrix}
\]

### 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.

Consider the set in R2: S2 = {(-1, 0), (-4, 1), (2, 3)}. Identify a basis for the span of this set.

Learn how to identify a basis for the span of a set of vectors in R^2. This problem demonstrates linear independence, row reduction, and finding a basis in vector spaces. Suitable for introductory linear algebra students.

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