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:

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.