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To find a basis for the given subspace, we need to examine the three vectors provided and remove any that are linearly dependent. The three vectors given are:

\[
v_1 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 5 \\ 0 \\ -1 \\ 0 \end{bmatrix}, \quad v_3 = \begin{bmatrix} 3 \\ 0 \\ 3 \\ 3 \end{bmatrix}
\]

The method to find the basis is as follows:

### Step 1: Form a matrix from the vectors.
Construct a matrix \( A \) using the given vectors as its columns:

\[
A = \begin{bmatrix}
0 & 5 & 3 \\
0 & 0 & 0 \\
0 & -1 & 3 \\
1 & 0 & 3
\end{bmatrix}
\]

### Step 2: Perform row reduction.
We will perform Gaussian elimination (row reduction) on the matrix \( A \) to determine if the vectors are linearly independent. If a row of all zeros appears during the reduction, it indicates linear dependence.

The row-reduced form of \( A \) (also called the row echelon form) is:

\[
\text{Row-reduced form of } A = \begin{bmatrix}
1 & 0 & 3 \\
0 & 1 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]

### Step 3: Identify linearly independent vectors.
From the row-reduced form, we can see that the first two columns contain pivots, meaning they correspond to linearly independent vectors. The third column is a linear combination of the first two, so it is dependent.

### Step 4: Basis for the subspace.
The linearly independent vectors are \( v_1 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \) and \( v_2 = \begin{bmatrix} 5 \\ 0 \\ -1 \\ 0 \end{bmatrix} \).

Thus, a basis for the subspace is:

\[
\left\{ \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 5 \\ 0 \\ -1 \\ 0 \end{bmatrix} \right\}
\]

Let me know if you would like more details or clarifications! 

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Here are 5 related questions that could deepen your understanding:

1. What is the general method to determine linear independence of vectors?
2. How can you apply Gaussian elimination to find the rank of a matrix?
3. Why is the concept of a "basis" important in vector spaces?
4. How do you interpret the dimension of a subspace based on its basis?
5. Can you explain how vector spaces are related to systems of linear equations?

### Tip:
When checking for linear dependence, remember that a set of vectors is dependent if one vector can be written as a linear combination of the others.

Find a basis of the given subspace by deleting linearly dependent vectors from the span of three vectors.

Learn how to find a basis for a subspace by removing linearly dependent vectors. This problem demonstrates Gaussian elimination and the basis theorem, suitable for undergraduate students studying linear algebra.

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