To solve this problem, we need to determine if the vectors in the set $\{ \begin{bmatrix}-6 \\ 2 \\ -2 \\ -2 \end{bmatrix}, \begin{bmatrix}-3 \\ 3 \\ 9 \\ -3 \end{bmatrix}, \begin{bmatrix}-9 \\ -5 \\ -3 \\ -5 \end{bmatrix} \}$ are linearly independent and, if not, extract a basis for the subspace $U \subset \mathbb{R}^4$.

Step-by-Step Solution:

1. Set Up the Matrix: Construct a matrix $A$ with these vectors as columns: $A = \begin{bmatrix} -6 & -3 & -9 \\ 2 & 3 & -5 \\ -2 & 9 & -3 \\ -2 & -3 & -5 \end{bmatrix}$

2. Row Reduction: Perform row reduction to find the row echelon form of $A$. The pivot columns will correspond to the linearly independent vectors that form a basis for $U$.

3. Determine the Basis: After finding the row echelon form, identify which vectors (columns) are linearly independent and thus form a basis for $U$.

4. Verification with Given Basis Options: The problem also suggests a possible basis in terms of a given vector set: $\left\{ \begin{bmatrix} 0 \\ 2 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} -3 \\ -1 \\ -3 \\ -1 \end{bmatrix}, x \right\}$ We need to determine the third vector $x$ to complete the basis. $x$ should be chosen such that the set remains linearly independent and spans the same subspace.

I'll calculate these steps to help determine the exact form of $x$.The row reduction process confirms that the rank of matrix $A$ is 3, meaning that the three vectors given are linearly independent and already span a three-dimensional subspace of $\mathbb{R}^4$. Therefore, these three vectors form a basis for $U$ as they are linearly independent.

Now, regarding the form of the vector $x$ in the basis representation given in the problem, it must also lie in the span of $U$. Since we have established that the three vectors provided are independent, any vector $x$ would need to be a linear combination of these vectors to be part of the basis for $U$.

If you need further clarification on choosing $x$ or additional steps, please let me know!

Here are five related questions you might consider next:

1. How can we determine if additional vectors are linearly dependent on a given set?
2. What is the significance of the rank of a matrix in terms of vector spaces?
3. How does the choice of a basis affect coordinates in a subspace?
4. Can we choose multiple different bases for the same subspace?
5. What does it mean for a set of vectors to "span" a space?

Tip: When constructing a basis, ensure that all vectors in the basis are linearly independent and cover the desired dimensionality of the subspace.