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To determine whether the sets of vectors are linearly independent, we need to check if the only solution to the equation \( c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + \dots + c_n \mathbf{v_n} = \mathbf{0} \) is the trivial solution (where all coefficients \( c_1, c_2, \dots, c_n = 0 \)).

Let's proceed with the given sets of vectors:

### Part A: Set \( S \)

\[
S = \left\{ \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ -2 \end{pmatrix}, \begin{pmatrix} 3 \\ -3 \\ 8 \end{pmatrix} \right\}
\]

We will form a matrix with these vectors as columns and row-reduce it to check if it has a pivot in every column (i.e., if the rank is equal to the number of vectors).

The matrix is:

\[
\begin{pmatrix}
2 & 1 & 3 \\
-1 & 1 & -3 \\
3 & -2 & 8
\end{pmatrix}
\]

### Part B: Set \( T \)

\[
T = \left\{ \begin{pmatrix} 1 \\ 2 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} \right\}
\]

The matrix formed from these vectors is:

\[
\begin{pmatrix}
1 & 1 & 1 \\
2 & 1 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 1 & 1
\end{pmatrix}
\]

Now, I will row-reduce these matrices to determine if the sets are linearly independent.### Results:

- **Set \( S \):**
  The matrix formed from the vectors of \( S \) has rank 2, which is less than the number of vectors (3). This means the vectors in set \( S \) are **linearly dependent**.

- **Set \( T \):**
  The matrix formed from the vectors of \( T \) has rank 3, which is equal to the number of vectors (3). This means the vectors in set \( T \) are **linearly independent**.

Would you like more details or have any questions?

Here are five questions you might find helpful:
1. What is the process for row-reducing a matrix manually?
2. How do you determine the rank of a matrix?
3. What does it mean for vectors to span a space?
4. How does the number of vectors relate to their independence?
5. What is the geometric interpretation of linear dependence?

**Tip:** Linear independence means no vector in the set can be written as a linear combination of the others.

Determine whether the following sets of vectors are linearly independent: A. S = {(2, -1, 3), (1, 1, -2), (3, -3, 8)}, B. T = {(1, 2, 1, 0, 0), (1, 1, 0, 1, 1), (1, 1, 0, 0, 1)}.

Explore how to determine whether the sets of vectors in S = {(2, -1, 3), (1, 1, -2), (3, -3, 8)} and T = {(1, 2, 1, 0, 0), (1, 1, 0, 1, 1), (1, 1, 0, 0, 1)} are linearly independent. This solution involves row-reduction and checking the rank of the matrix formed by the vectors. Suitable for undergraduate students learning linear algebra.

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