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:

   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:

   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.