s the set 
{
(
3
6
9
)
,
 
(
15
30
45
)
,
 
(
−
5
−
6
−
15
)
}
⎩
⎨
⎧
​
  
​
  
3
6
9
​
  
​
 ,  
​
  
15
30
45
​
  
​
 ,  
​
  
−5
−6
−15
​
  
​
  
⎭
⎬
⎫
​
  linearly independent?

To determine if the set 
\[
\left\{
\begin{pmatrix} 3 \\ 6 \\ 9 \end{pmatrix}, \begin{pmatrix} 15 \\ 30 \\ 45 \end{pmatrix}, \begin{pmatrix} -5 \\ -6 \\ -15 \end{pmatrix} \right\}
\]
is linearly independent, we need to check if there exist scalars \( c_1 \), \( c_2 \), and \( c_3 \), not all zero, such that:

\[
c_1 \begin{pmatrix} 3 \\ 6 \\ 9 \end{pmatrix} + c_2 \begin{pmatrix} 15 \\ 30 \\ 45 \end{pmatrix} + c_3 \begin{pmatrix} -5 \\ -6 \\ -15 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}.
\]

This equation translates into a system of linear equations:

\[
\begin{cases}
3c_1 + 15c_2 - 5c_3 = 0, \\
6c_1 + 30c_2 - 6c_3 = 0, \\
9c_1 + 45c_2 - 15c_3 = 0.
\end{cases}
\]

### Step 1: Set up the augmented matrix
The augmented matrix for this system is:

\[
\begin{pmatrix}
3 & 15 & -5 & | & 0 \\
6 & 30 & -6 & | & 0 \\
9 & 45 & -15 & | & 0 \\
\end{pmatrix}
\]

### Step 2: Row reduce the matrix
We can perform row operations to simplify this matrix and determine if there is a non-trivial solution (a solution where at least one of the \( c_i \)'s is not zero).

1. Divide the first row by 3 to simplify:

   \[
   \begin{pmatrix}
   1 & 5 & -\frac{5}{3} & | & 0 \\
   6 & 30 & -6 & | & 0 \\
   9 & 45 & -15 & | & 0 \\
   \end{pmatrix}
   \]

2. Subtract 6 times the first row from the second row and 9 times the first row from the third row:

   \[
   \begin{pmatrix}
   1 & 5 & -\frac{5}{3} & | & 0 \\
   0 & 0 & 4 & | & 0 \\
   0 & 0 & 0 & | & 0 \\
   \end{pmatrix}
   \]

From this row-reduced form, we see that the second row simplifies to \( 0 = 0 \), which indicates a dependency among the vectors. Specifically, the matrix has only two pivot positions (one in the first column and one in the third column), suggesting that the rank of the matrix is less than 3.

### Conclusion
Since the rank is less than the number of vectors, these vectors are **linearly dependent**. Therefore, the set is **not linearly independent**.

Would you like further details or have questions about this solution?

Here are 5 related questions for further exploration:
1. How does one determine linear independence in higher dimensions?
2. Can a set of two vectors in \( \mathbb{R}^3 \) be linearly independent?
3. How is linear dependence related to the concept of basis?
4. What is the significance of the rank of a matrix in terms of vector spaces?
5. How does one find a basis for the span of a set of dependent vectors?

**Tip:** Linear dependence among vectors implies that one vector can be expressed as a combination of others.

Is the set { (3, 6, 9), (15, 30, 45), (-5, -6, -15) } linearly independent?

Determine if the set of vectors {(3, 6, 9), (15, 30, 45), (-5, -6, -15)} is linearly independent using matrix row reduction. This solution explores key concepts in linear algebra, including linear independence, vector spaces, and rank determination. Suitable for Grades 11-12 and undergraduate students.

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