Check whether the following set of vectors is LD or LI ?S3 = {(1, 5, 2),(0, 0, 1),(1, 1, 0)}

To check if the set of vectors \( S_3 = \{ (1, 5, 2), (0, 0, 1), (1, 1, 0) \} \) is linearly dependent (LD) or linearly independent (LI), we will perform the following steps:

### Step 1: Write the vectors as columns of a matrix
We will represent the vectors as columns of a matrix:

\[
A = \begin{pmatrix}
1 & 0 & 1 \\
5 & 0 & 1 \\
2 & 1 & 0
\end{pmatrix}
\]

### Step 2: Perform Gaussian elimination
We now perform row operations to reduce the matrix to row echelon form.

1. The first pivot element is \( 1 \) in row 1, column 1. We keep this as is.
2. Subtract 5 times the first row from the second row to eliminate the second row's first entry:
   \[
   R_2 = R_2 - 5R_1 = (5, 0, 1) - 5(1, 0, 2) = (0, 0, -9)
   \]
   Now the matrix is:
   \[
   \begin{pmatrix}
   1 & 0 & 1 \\
   0 & 0 & -9 \\
   2 & 1 & 0
   \end{pmatrix}
   \]

3. Subtract 2 times the first row from the third row:
   \[
   R_3 = R_3 - 2R_1 = (2, 1, 0) - 2(1, 0, 2) = (0, 1, -4)
   \]
   Now the matrix is:
   \[
   \begin{pmatrix}
   1 & 0 & 1 \\
   0 & 0 & -9 \\
   0 & 1 & -4
   \end{pmatrix}
   \]

4. Divide the second row by \(-9\) to make the pivot entry \( 1 \):
   \[
   R_2 = \frac{1}{-9}R_2 = (0, 0, 1)
   \]
   Now the matrix is:
   \[
   \begin{pmatrix}
   1 & 0 & 1 \\
   0 & 0 & 1 \\
   0 & 1 & -4
   \end{pmatrix}
   \]

5. Subtract \( 1 \) times the second row from the first row:
   \[
   R_1 = R_1 - R_2 = (1, 0, 1) - (0, 0, 1) = (1, 0, 0)
   \]
   The matrix becomes:
   \[
   \begin{pmatrix}
   1 & 0 & 0 \\
   0 & 0 & 1 \\
   0 & 1 & -4
   \end{pmatrix}
   \]

### Step 3: Conclusion
The matrix has 3 pivot columns, meaning the rank of the matrix is 3. Since the number of pivot columns equals the number of vectors, the set of vectors is **linearly independent (LI)**.

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Would you like more details on Gaussian elimination or the linear dependence test? Here are some related questions:

1. What does it mean for vectors to be linearly independent?
2. How do we use determinants to check linear independence?
3. Can a set of two vectors be linearly dependent in \( \mathbb{R}^3 \)?
4. How does the rank of a matrix relate to the number of linearly independent vectors?
5. Can you show an example where vectors are linearly dependent?

**Tip:** When checking linear dependence, always ensure that the number of pivot columns matches the number of vectors to confirm linear independence.

Check whether the following set of vectors is LD or LI? S3 = {(1, 5, 2), (0, 0, 1), (1, 1, 0)}

Check the linear dependence or independence of the vector set S3 = {(1, 5, 2), (0, 0, 1), (1, 1, 0)} using Gaussian elimination. Understand how matrix representation and rank help in determining whether a set of vectors is LD or LI. Suitable for college-level linear algebra students.

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