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Part 1
Let Bold v 1equalsleft bracket Start 3 By 1 Matrix 1st Row 1st Column 0 2nd Row 1st Column 0 3rd Row 1st Column negative 2 EndMatrix right bracket
​, Bold v 2equalsleft bracket Start 3 By 1 Matrix 1st Row 1st Column 0 2nd Row 1st Column negative 4 3rd Row 1st Column negative 4 EndMatrix right bracket
​, and Bold v 3equalsleft bracket Start 3 By 1 Matrix 1st Row 1st Column 6 2nd Row 1st Column negative 5 3rd Row 1st Column negative 6 EndMatrix right bracket
. Does StartSet Bold v 1 comma Bold v 2 comma Bold v 3 EndSet span set of real numbers R cubed​? Why or why​ not?

To determine whether the set \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\) spans \(\mathbb{R}^3\), we need to check if these vectors are linearly independent and if they form a basis for \(\mathbb{R}^3\).

The vectors are:

\[
\mathbf{v}_1 = \begin{bmatrix} 0 \\ 0 \\ -2 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 0 \\ -4 \\ -4 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 6 \\ -5 \\ -6 \end{bmatrix}
\]

### Step 1: Form the matrix
We construct a matrix by placing \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) as the columns of a 3x3 matrix:

\[
A = \begin{bmatrix}
0 & 0 & 6 \\
0 & -4 & -5 \\
-2 & -4 & -6
\end{bmatrix}
\]

### Step 2: Row reduce the matrix
We need to row-reduce this matrix to determine if the vectors are linearly independent. If the matrix has full rank (rank 3), then the vectors span \(\mathbb{R}^3\).

1. The first column is already simplified since the first two entries are zero.
2. The third row is already in reduced form for the first column.
3. Now, work on reducing the second and third columns.

I will perform these calculations and row-reduce the matrix to check the rank.The row-reduced form of the matrix is the identity matrix:

\[
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\]

This indicates that the matrix has full rank (rank 3), meaning that the vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) are linearly independent.

### Conclusion:
Since the vectors are linearly independent, they span \(\mathbb{R}^3\). Therefore, the set \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}\) **does** span \(\mathbb{R}^3\).

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Would you like more details or have any further questions? Here are five related questions to consider:

1. What is the geometric interpretation of spanning \(\mathbb{R}^3\)?
2. How can you check if two vectors span a plane in \(\mathbb{R}^3\)?
3. Can a set of dependent vectors ever span \(\mathbb{R}^3\)?
4. How does the rank of a matrix relate to the number of vectors that span a space?
5. What happens if one of the vectors is the zero vector in terms of spanning?

**Tip:** Linear independence is crucial for determining the span of vectors in a space—always check this property by analyzing the matrix's rank.

Does the set {v1, v2, v3} span the set of real numbers R^3? Why or why not? Where v1 = [0, 0, -2], v2 = [0, -4, -4], v3 = [6, -5, -6].

Explore how to determine whether the set of vectors {v1, v2, v3} spans R^3 using matrix row reduction and linear independence. Understand the concepts of vector spaces, rank, and span in linear algebra. Suitable for undergraduate students.

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