Determine if  sets of vectors is linearly independent or linearly dependent.
 V = {[1 0 1], [1 2 1], [2 2 2]}


To determine if the set of vectors \( V = \{[1, 0, 1], [1, 2, 1], [2, 2, 2]\} \) is linearly independent or linearly dependent, we need to check if the only solution to the equation

\[
c_1 \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + c_2 \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} + c_3 \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}
\]

is \( c_1 = c_2 = c_3 = 0 \).

This involves setting up the following matrix equation and determining if the matrix has a non-trivial solution:

\[
\begin{bmatrix}
1 & 1 & 2 \\
0 & 2 & 2 \\
1 & 1 & 2
\end{bmatrix}
\begin{bmatrix}
c_1 \\
c_2 \\
c_3
\end{bmatrix}
=
\begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix}
\]

To solve this, we can row-reduce the matrix to its reduced row echelon form (RREF) and see if there are any free variables.

Let's proceed with the row reduction.

### Step 1: Set up the augmented matrix:
\[
\begin{bmatrix}
1 & 1 & 2 & | & 0 \\
0 & 2 & 2 & | & 0 \\
1 & 1 & 2 & | & 0
\end{bmatrix}
\]

### Step 2: Perform row operations to achieve RREF.
- Subtract the first row from the third row to eliminate the first element in the third row:
\[
\begin{bmatrix}
1 & 1 & 2 & | & 0 \\
0 & 2 & 2 & | & 0 \\
0 & 0 & 0 & | & 0
\end{bmatrix}
\]
- Divide the second row by 2 to simplify:
\[
\begin{bmatrix}
1 & 1 & 2 & | & 0 \\
0 & 1 & 1 & | & 0 \\
0 & 0 & 0 & | & 0
\end{bmatrix}
\]
- Subtract the second row from the first row to eliminate the second element in the first row:
\[
\begin{bmatrix}
1 & 0 & 1 & | & 0 \\
0 & 1 & 1 & | & 0 \\
0 & 0 & 0 & | & 0
\end{bmatrix}
\]

### Step 3: Analyze the result.
The RREF matrix shows that there is a free variable (associated with the third column), meaning that there is a non-trivial solution to the equation.

Since the system has a non-trivial solution, the vectors are **linearly dependent**.

Would you like more details, or do you have any questions?

Here are some related questions you might consider:
1. What is the geometric interpretation of linearly dependent vectors?
2. How would the analysis change if a different set of vectors was given?
3. Can a set of three vectors in \(\mathbb{R}^3\) ever be linearly independent?
4. What role do row operations play in determining linear dependence?
5. How does the determinant of the matrix relate to the independence of the vectors?

**Tip:** To check for linear dependence in small dimensions, you can also compute the determinant of the matrix formed by the vectors. If the determinant is zero, the vectors are linearly dependent.

Learn how to determine if the set of vectors V = {[1 0 1], [1 2 1], [2 2 2]} is linearly independent or dependent. Step-by-step solution involving matrix equation solving and row reduction. Suitable for undergraduate students studying linear algebra.

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