Extend B = {(1,2,-1,1),(0,1,2,-1)} to a basis of R^4

We are tasked with extending the set \( B = \{(1, 2, -1, 1), (0, 1, 2, -1)\} \) to form a basis for \( \mathbb{R}^4 \).

### Step 1: Verify if the set is linearly independent
The set \( B \) consists of two vectors in \( \mathbb{R}^4 \). To extend this set into a basis for \( \mathbb{R}^4 \), we first need to check whether these two vectors are linearly independent.

Let:
\[
\mathbf{v}_1 = (1, 2, -1, 1), \quad \mathbf{v}_2 = (0, 1, 2, -1)
\]
To check for linear independence, we form the matrix with \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) as its columns and compute the rank:
\[
A = \begin{pmatrix}
1 & 0 \\
2 & 1 \\
-1 & 2 \\
1 & -1
\end{pmatrix}
\]
By row reducing this matrix, we can determine whether the vectors are linearly independent.

### Step 2: Row reduce the matrix
We perform Gaussian elimination on the matrix \( A \):
\[
\begin{pmatrix}
1 & 0 \\
2 & 1 \\
-1 & 2 \\
1 & -1
\end{pmatrix}
\]
First, subtract \( 2 \times \text{Row 1} \) from Row 2:
\[
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
-1 & 2 \\
1 & -1
\end{pmatrix}
\]
Next, add Row 1 to Row 3, and subtract Row 1 from Row 4:
\[
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
0 & 2 \\
0 & -1
\end{pmatrix}
\]
Finally, divide Row 3 by 2 and multiply Row 4 by \(-1\):
\[
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1
\end{pmatrix}
\]
Subtract Row 3 from Row 4:
\[
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 0
\end{pmatrix}
\]
So, the rank of the matrix is 2, confirming that \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) are linearly independent.

### Step 3: Extend to a basis of \( \mathbb{R}^4 \)
Since \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) form a linearly independent set in \( \mathbb{R}^4 \), we need to find two more linearly independent vectors to form a basis for \( \mathbb{R}^4 \).

We'll choose two additional vectors from \( \mathbb{R}^4 \) that are linearly independent of \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \). One straightforward choice is to take the standard basis vectors \( \mathbf{e}_3 = (0, 0, 1, 0) \) and \( \mathbf{e}_4 = (0, 0, 0, 1) \).

### Step 4: Verify independence of the new set
The set of vectors we propose is:
\[
B' = \left\{(1, 2, -1, 1), (0, 1, 2, -1), (0, 0, 1, 0), (0, 0, 0, 1)\right\}
\]
We now verify if these four vectors are linearly independent. To do this, we can form a matrix with these vectors as columns:
\[
A' = \begin{pmatrix}
1 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 \\
-1 & 2 & 1 & 0 \\
1 & -1 & 0 & 1
\end{pmatrix}
\]
This matrix is already in row echelon form, and all columns are pivot columns, meaning that the vectors are linearly independent.

### Step 5: Conclusion
The set \( B' = \{(1, 2, -1, 1), (0, 1, 2, -1), (0, 0, 1, 0), (0, 0, 0, 1)\} \) is a basis for \( \mathbb{R}^4 \).

---

Would you like a detailed breakdown of why these vectors were chosen, or do you have any other questions?

### Further questions:
1. How do you identify linearly dependent vectors in a set?
2. What happens if you choose dependent vectors when extending a basis?
3. Can any two independent vectors in \( \mathbb{R}^4 \) be extended to a basis?
4. What are the properties of a basis in a vector space?
5. How do row operations help in identifying linear independence?

### Tip:
When extending a set to form a basis, it's often useful to add standard basis vectors if they're linearly independent from the existing set.

Learn how to extend the set B = {(1,2,-1,1),(0,1,2,-1)} to form a basis for R^4 using linear algebra concepts such as vector spaces, linear independence, and matrix row reduction. Suitable for college-level students studying Linear Algebra.

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