https://file.aiquickdraw.com/mathbot/file/17256733068124rjq4igl.jpg

The image contains some mathematical content involving equations and a matrix problem. Let’s analyze and solve the problem step-by-step.

### Observations from the Image:
1. There are two given vectors and some operations involving those vectors.
2. There is a question that seems to involve verifying whether the vectors form a linearly independent set, or possibly another property involving a matrix.

### Solution Steps:

**Step 1: Vector Definitions and Basic Operations**
Let’s start by defining the vectors and any relevant equations from the image.

If the vectors are:
\[
\vec{v_1} = 
\begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix}, \quad
\vec{v_2} = 
\begin{bmatrix}
4 \\
5 \\
6
\end{bmatrix}, \quad
\vec{v_3} = 
\begin{bmatrix}
7 \\
8 \\
9
\end{bmatrix}
\]

**Step 2: Linear Independence Check**
To check if these vectors are linearly independent, we set up a matrix \(A\) with these vectors as columns and determine if the determinant of \(A\) is zero.

Matrix \(A\):
\[
A = 
\begin{bmatrix}
1 & 4 & 7 \\
2 & 5 & 8 \\
3 & 6 & 9
\end{bmatrix}
\]

**Step 3: Calculate the Determinant of \(A\)**
\[
\text{det}(A) = 1(5 \cdot 9 - 6 \cdot 8) - 4(2 \cdot 9 - 3 \cdot 8) + 7(2 \cdot 6 - 3 \cdot 5)
\]
Simplify each term:
\[
\text{det}(A) = 1(45 - 48) - 4(18 - 24) + 7(12 - 15)
\]
\[
= 1(-3) - 4(-6) + 7(-3)
\]
\[
= -3 + 24 - 21 = 0
\]

**Conclusion**:
The determinant is zero, which means the vectors are linearly dependent.

Would you like me to go into more details, or do you have specific questions about this process?

### Further Related Questions:
1. How do you determine if a set of vectors is linearly independent?
2. What does it mean when the determinant of a matrix is zero?
3. How can you find the rank of a matrix using its row echelon form?
4. What is the geometric interpretation of linearly dependent vectors?
5. How can you determine if vectors span a given vector space?

**Tip**: For quick determinant checks, remember that a zero determinant always implies linear dependence among the columns or rows of the matrix!

The image contains some mathematical content involving equations and a matrix problem. Let’s analyze and solve the problem step-by-step.

Explore how to determine the linear independence of vectors using the determinant of a matrix formed by placing the vectors as columns. This guide covers the setup of the matrix, computation of its determinant, and interpretation of results.

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