Given the vectors (0,−1,2,1),(a,−1,0,1),(1,−1,b,0) calculate a and b so they generate a subspace of dimension 2. Please do it with the row reducing method by putting each vector in a column

To find values of \( a \) and \( b \) such that the vectors \((0, -1, 2, 1)\), \((a, -1, 0, 1)\), and \((1, -1, b, 0)\) generate a subspace of dimension 2, we will arrange these vectors as columns in a matrix and then use row reduction to determine conditions on \( a \) and \( b \).

### Step 1: Form the matrix

The matrix formed by the vectors as columns is:

\[
A = \begin{bmatrix}
0 & a & 1 \\
-1 & -1 & -1 \\
2 & 0 & b \\
1 & 1 & 0
\end{bmatrix}
\]

### Step 2: Row reduce the matrix

We'll perform row operations to row reduce this matrix.

1. Swap the first row with the second row to get a leading 1:

\[
\begin{bmatrix}
-1 & -1 & -1 \\
0 & a & 1 \\
2 & 0 & b \\
1 & 1 & 0
\end{bmatrix}
\]

2. Multiply the first row by \(-1\):

\[
\begin{bmatrix}
1 & 1 & 1 \\
0 & a & 1 \\
2 & 0 & b \\
1 & 1 & 0
\end{bmatrix}
\]

3. Eliminate the first entry of rows 3 and 4:

- Row 3: \( R_3 - 2R_1 \)
- Row 4: \( R_4 - R_1 \)

This gives:

\[
\begin{bmatrix}
1 & 1 & 1 \\
0 & a & 1 \\
0 & -2 & b - 2 \\
0 & 0 & -1
\end{bmatrix}
\]

4. Now, continue to row reduce:

- For Row 3, replace \( R_3 \) with \( R_3 + 2R_2 \):

\[
\begin{bmatrix}
1 & 1 & 1 \\
0 & a & 1 \\
0 & 0 & b - 2 + 2a \\
0 & 0 & -1
\end{bmatrix}
\]

### Step 3: Find conditions for dimension 2

To ensure that the rank of the matrix is 2, the third row must become a zero row. Thus, we require:

\[
b - 2 + 2a = 0
\]

From this, we can express \( b \) in terms of \( a \):

\[
b = 2 - 2a
\]

### Step 4: Set a specific value for \( a \)

We can choose any value for \( a \) to generate corresponding \( b \). For instance, if we let \( a = 1 \):

\[
b = 2 - 2(1) = 0
\]

### Conclusion

The values \( a = 1 \) and \( b = 0 \) will ensure that the vectors span a subspace of dimension 2.

### Summary of Results
- \( a = 1 \)
- \( b = 0 \)

Do you need further details or explanations? Here are some related questions:
1. How does the dimension of a subspace relate to the rank of a matrix?
2. What are the implications of having linearly dependent vectors?
3. Can you provide an example of how to visualize these vectors in a 4D space?
4. What are other possible pairs of \( (a, b) \) that satisfy the condition?
5. How can this method be applied to a different set of vectors? 

**Tip:** Always check the linear independence of vectors by forming a matrix and reducing it to row echelon form.

Given the vectors (0,−1,2,1),(a,−1,0,1),(1,−1,b,0) calculate a and b so they generate a subspace of dimension 2. Please do it with the row reducing method by putting each vector in a column.

Explore how to calculate values of a and b so that vectors (0,−1,2,1), (a,−1,0,1), and (1,−1,b,0) form a subspace of dimension 2. This solution covers linear independence, matrix row reduction, and the rank-nullity theorem, tailored for linear algebra students.

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