https://file.aiquickdraw.com/mathbot/file/1725248583050brx5uuc2.jpg

To solve the problem using the determinant method, we first need to confirm the matrix \(B\) for Option (b) from the image.

### Option (b) Matrix

From the image, the vectors for option (b) are:

\[
\begin{pmatrix}
5 \\ 2 \\ 0 \\ 1
\end{pmatrix},
\begin{pmatrix}
1 \\ 0 \\ -3 \\ 0
\end{pmatrix},
\begin{pmatrix}
1 \\ 0 \\ -2 \\ 2
\end{pmatrix},
\begin{pmatrix}
3 \\ 1 \\ 0 \\ 1
\end{pmatrix}
\]

These vectors form a 4x4 matrix \(B\):

\[
B = \begin{pmatrix}
5 & 1 & 1 & 3 \\
2 & 0 & 0 & 1 \\
0 & -3 & -2 & 0 \\
1 & 0 & 2 & 1
\end{pmatrix}
\]

### Determinant Method

To determine if the vectors are linearly dependent, we calculate the determinant of matrix \(B\). If the determinant is zero, the vectors are linearly dependent. If it's non-zero, they are linearly independent.

Let me calculate the determinant of matrix \(B\).The determinant of matrix \(B\) is approximately \(5\). 

Since the determinant is not zero, the set of vectors in Option (b) is **linearly independent**.

Would you like more details or have any questions?

### Further Questions:
1. What does it mean for a set of vectors to be linearly independent in a vector space?
2. How can we visually interpret linear dependence or independence in \(\mathbb{R}^2\) or \(\mathbb{R}^3\)?
3. What are some real-world applications where determining linear dependence is crucial?
4. How would the result change if we altered one of the vectors slightly?
5. Can linear independence be determined by any other methods apart from the determinant?

### Tip:
When dealing with determinants, it's crucial to ensure that you perform the calculations carefully, as even small errors can lead to incorrect conclusions about linear dependence or independence.

Learn how to determine linear dependence using the determinant method with a detailed explanation. This problem involves calculating the determinant of a matrix to ascertain linear dependence or independence of vectors in a vector space. Suitable for college-level students studying linear algebra.

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