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

To determine the values of \( a \) and \( b \) such that the vectors \( \mathbf{v_1} = (0, -1, 2, 1) \), \( \mathbf{v_2} = (a, -1, 0, 1) \), and \( \mathbf{v_3} = (1, -1, b, 0) \) generate a subspace of dimension 2, we need to ensure that at least one vector can be expressed as a linear combination of the others, indicating linear dependence among them.

### Steps to solve:

1. **Form the Matrix**:
   We can arrange the vectors as rows in a matrix and perform row operations to determine conditions for linear dependence:

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

2. **Row Reduction**:
   We will row-reduce this matrix to check for dependencies.

   Start by interchanging the first and third rows:

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

   Then, eliminate the first column below the pivot:

   - Row 2: \( R_2 - a \cdot R_1 \)
   - Row 3 remains the same.

   The matrix now looks like:

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

3. **Further Reduction**:
   We simplify Row 2 and Row 3. Subtract Row 3 from Row 2 to find conditions for linear dependence.

   After performing row operations, we aim for a zero row indicating linear dependence, which leads us to:

   \[
   (-1 + a) + 1 = 0 \quad \text{and} \quad -ab + 2 = 0
   \]

4. **Set Up Equations**:
   From the conditions derived:

   1. \( a - 1 = 0 \) → \( a = 1 \)
   2. \( -ab + 2 = 0 \)

   Substitute \( a = 1 \) into the second equation:

   \[
   -1 \cdot b + 2 = 0 \quad \Rightarrow \quad b = 2
   \]

### Conclusion:
Thus, the values are:

\[
\boxed{a = 1, \, b = 2}
\]

These values ensure the three vectors span a subspace of dimension 2.

### Further Exploration:
1. How can you verify that the vectors are linearly dependent with these values?
2. What would happen if \( a \) and \( b \) were different?
3. Can you find another set of values for \( a \) and \( b \) that also leads to a dimension of 2?
4. What is the geometric interpretation of these vectors in \( \mathbb{R}^4 \)?
5. How does the choice of vectors affect the dimension of the generated subspace?

**Tip**: Always check for linear dependence by forming a matrix and reducing it to see if you can obtain a row of zeros.

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.

Explore how to calculate values a and b for vectors (0,−1,2,1), (a,−1,0,1), and (1,−1,b,0) to form a subspace of dimension 2. This guide includes a step-by-step solution involving row reduction and linear dependence, suitable for high school students.

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