Are the vectors [−24],[7−2]
and [3−6]
linearly independent?
Yes
No
No, the answer is incorrect.

To determine whether the vectors \( \mathbf{v_1} = \begin{bmatrix} -2 \\ 4 \end{bmatrix}, \mathbf{v_2} = \begin{bmatrix} 7 \\ -2 \end{bmatrix}, \mathbf{v_3} = \begin{bmatrix} 3 \\ -6 \end{bmatrix} \) are linearly independent, we check if the equation:

\[
c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + c_3 \mathbf{v_3} = \mathbf{0}
\]

has only the trivial solution (where all the constants \(c_1 = c_2 = c_3 = 0\)).

This leads to the system of equations:
\[
c_1(-2) + c_2(7) + c_3(3) = 0 \quad \text{(from the first row)}
\]
\[
c_1(4) + c_2(-2) + c_3(-6) = 0 \quad \text{(from the second row)}
\]

Let’s solve this system to determine if there is a nontrivial solution or not.

### Step 1: First equation
\[
-2c_1 + 7c_2 + 3c_3 = 0
\]

### Step 2: Second equation
\[
4c_1 - 2c_2 - 6c_3 = 0
\]

We now solve this system to check for linear independence.

The solution to the system is:

\[
c_1 = \frac{3}{2} c_3, \quad c_2 = 0
\]

This shows that \(c_1\) and \(c_3\) are dependent on each other, and there is a non-trivial solution when \(c_3 \neq 0\). Therefore, the vectors are **linearly dependent**, not independent.

Let me know if you would like further clarification or details!

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Here are 5 related questions to expand on this:
1. What are the implications of vectors being linearly dependent?
2. Can you explain how the number of vectors affects linear independence in higher dimensions?
3. How do you use matrix methods (like determinants) to test for linear independence?
4. What are the geometric interpretations of linearly dependent vectors?
5. How would this change if the vectors were in 3D space?

**Tip**: When testing linear independence, you can quickly check if the determinant of the matrix formed by the vectors (in the case of square matrices) is zero. A zero determinant means the vectors are dependent.

Are the vectors [-2 4], [7 -2] and [3 -6] linearly independent?

This page provides a step-by-step solution to determine if the vectors [-2 4], [7 -2], and [3 -6] are linearly independent. It explores the concepts of vector spaces, linear dependence, and independence, solving the system of equations to reveal whether a non-trivial solution exists. Suitable for advanced high school or undergraduate students.

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