Determine whether (-11,-1,1), (1,2,3,2), (2,5,6,4) and (-2,6,8,5) spans R*.If not find the dimension of the subspace they span.

To determine whether the vectors \((-11, -1, 1)\), \((1, 2, 3, 2)\), \((2, 5, 6, 4)\), and \((-2, 6, 8, 5)\) span \(\mathbb{R}^*\) (which I assume means \(\mathbb{R}^4\), the 4-dimensional real vector space), we need to analyze their linear independence. If these vectors are linearly independent, then they span \(\mathbb{R}^4\). If they are not, we must find the dimension of the subspace they span.

### Step 1: Arrange the Vectors as a Matrix

We first arrange the given vectors as rows in a matrix \(A\):

\[
A = \begin{pmatrix}
-11 & -1 & 1 & 0 \\
1 & 2 & 3 & 2 \\
2 & 5 & 6 & 4 \\
-2 & 6 & 8 & 5
\end{pmatrix}
\]

### Step 2: Perform Row Reduction

We'll perform row reduction (Gaussian elimination) to bring this matrix to row echelon form (REF) or reduced row echelon form (RREF). The number of non-zero rows in the REF or RREF will give us the rank of the matrix, which corresponds to the dimension of the span of these vectors.

Let's do this step by step.

\[
\text{Step 1: } R_1 \leftrightarrow R_4 \text{ to avoid fractions in first row.}
\]

\[
\begin{pmatrix}
-2 & 6 & 8 & 5 \\
1 & 2 & 3 & 2 \\
2 & 5 & 6 & 4 \\
-11 & -1 & 1 & 0 \\
\end{pmatrix}
\]

\[
\text{Step 2: } R_2 \rightarrow R_2 + \frac{1}{2} R_1, \quad R_3 \rightarrow R_3 - R_1, \quad R_4 \rightarrow R_4 + \frac{11}{2} R_1
\]

\[
\begin{pmatrix}
-2 & 6 & 8 & 5 \\
0 & 5 & 7 & \frac{9}{2} \\
0 & -7 & -10 & -6.5 \\
0 & 32 & 45 & 27.5
\end{pmatrix}
\]

\[
\text{Step 3: } R_3 \rightarrow R_3 + \frac{7}{5} R_2, \quad R_4 \rightarrow R_4 - \frac{32}{5} R_2
\]

\[
\begin{pmatrix}
-2 & 6 & 8 & 5 \\
0 & 5 & 7 & \frac{9}{2} \\
0 & 0 & 0.2 & -0.2 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

\[
\text{Step 4: } R_3 \rightarrow 5 R_3 
\]

\[
\begin{pmatrix}
-2 & 6 & 8 & 5 \\
0 & 5 & 7 & \frac{9}{2} \\
0 & 0 & 1 & -1 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

### Step 3: Determine the Rank

The resulting matrix has 3 non-zero rows, so the rank of this matrix is 3. This implies that the vectors span a subspace of \(\mathbb{R}^4\) of dimension 3.

### Conclusion

The vectors \((-11, -1, 1)\), \((1, 2, 3, 2)\), \((2, 5, 6, 4)\), and \((-2, 6, 8, 5)\) **do not** span \(\mathbb{R}^4\). Instead, they span a subspace of dimension 3.

Do you want details or have any questions?

Here are 8 related questions you might find useful:
1. How do you determine if vectors are linearly independent?
2. What is the significance of the rank of a matrix?
3. Can the span of fewer than 4 vectors in \(\mathbb{R}^4\) ever be \(\mathbb{R}^4\)?
4. How do you find a basis for the span of a set of vectors?
5. What are the properties of row echelon form versus reduced row echelon form?
6. Can the dimension of the span change if vectors are scaled or combined?
7. How does the concept of linear dependence relate to the span of vectors?
8. What is the relationship between the nullity and rank of a matrix?

**Tip:** When working with vector spaces, understanding the rank of a matrix can often simplify complex problems by giving insights into the dimensionality of the space involved.

Explore how to determine whether the vectors (-11, -1, 1), (1, 2, 3, 2), (2, 5, 6, 4), and (-2, 6, 8, 5) span R^4. Understand the concept of linear independence, perform matrix operations to find the rank, and discover the dimension of the subspace they span. Suitable for undergraduate students studying linear algebra.

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