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We are given the subspace \( W \) of \( \mathbb{R}^4 \), which is the span of the following vectors:

\[
W = \text{span} \left\{ 
\begin{bmatrix} 
1 \\ 
0 \\ 
0 \\ 
-1 
\end{bmatrix}, 
\begin{bmatrix} 
-4 \\ 
1 \\ 
-2 \\ 
2 
\end{bmatrix} 
\right\}.
\]

### Finding a Basis for \( W \):
To find a basis, we need to check if these two vectors are linearly independent. If they are, they form a basis of \( W \). Let's put these vectors into a matrix:

\[
A = \begin{bmatrix}
1 & -4 \\
0 & 1 \\
0 & -2 \\
-1 & 2
\end{bmatrix}
\]

We will perform row reduction on this matrix to check for linear independence.

1. The matrix is already in row echelon form since there are no further row operations needed to make it simpler.
   
\[
\text{RREF of } A = \begin{bmatrix}
1 & -4 \\
0 & 1 \\
0 & -2 \\
-1 & 2
\end{bmatrix}
\]

Since there are two pivot columns, this confirms that the two vectors are linearly independent.

Thus, the given two vectors form a **basis** for \( W \).

### Finding the Dimension of \( W \):
The dimension of a subspace is the number of vectors in its basis. Since we have found that the two given vectors are linearly independent, the **dimension** of \( W \) is:

\[
\boxed{2}
\]

Thus, the dimension of \( W \) is 2, and its basis is:

\[
\left\{
\begin{bmatrix} 
1 \\ 
0 \\ 
0 \\ 
-1 
\end{bmatrix}, 
\begin{bmatrix} 
-4 \\ 
1 \\ 
-2 \\ 
2 
\end{bmatrix} 
\right\}
\]

Would you like further details or explanations?

### Related Questions:
1. What is the rank of the matrix formed by these vectors?
2. How do we determine linear dependence or independence of more than two vectors?
3. How does the Gram-Schmidt process relate to finding a basis?
4. What is the geometric interpretation of the dimension of a subspace?
5. How would the process change if more vectors were given?

**Tip:** A set of vectors is linearly independent if no vector can be written as a linear combination of the others.

Consider the following subspace W of R^4: W = span{[1, 0, 0, -1], [-4, 1, -2, 2]}. Explain and demonstrate how to find a basis of W. Explain and demonstrate how to find the dimension of W.

Explore how to determine the basis and dimension of a subspace in R^4. This step-by-step guide demonstrates the process of checking for linear independence and finding a basis using matrix row reduction. Learn core concepts in linear algebra, such as vector spaces and the dimension theorem.

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