The image presents two vectors:

$\mathbf{v}_1 = \begin{bmatrix} 3 \\ 5 \\ 2 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 5 \\ 1 \\ -3 \end{bmatrix}$

The question is whether this set of vectors is linearly independent.

Determining Linear Independence:

Two vectors are linearly independent if the only solution to the equation:

$c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 = 0$

is $c_1 = c_2 = 0$. If there exist non-zero scalars $c_1$ and $c_2$ such that the above equation holds, the vectors are linearly dependent.

We can check this by setting up the following system of equations:

$c_1 \begin{bmatrix} 3 \\ 5 \\ 2 \end{bmatrix} + c_2 \begin{bmatrix} 5 \\ 1 \\ -3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$

This gives the system:

1. $3c_1 + 5c_2 = 0$
2. $5c_1 + 1c_2 = 0$
3. $2c_1 - 3c_2 = 0$

Solving this system can show whether the only solution is the trivial one ($c_1 = 0, c_2 = 0$).

Solving the system:

From equation (2):

$c_2 = -5c_1$

Substitute this into equations (1) and (3):

1. $3c_1 + 5(-5c_1) = 0$ $\Rightarrow 3c_1 - 25c_1 = 0 \Rightarrow -22c_1 = 0 \Rightarrow c_1 = 0$
2. Using $c_1 = 0$ in $c_2 = -5c_1$, we get $c_2 = 0$.

Thus, the only solution is $c_1 = 0$ and $c_2 = 0$, meaning the vectors are linearly independent.

Conclusion:

The statement in the image is False because the set of vectors is linearly independent. Therefore, the correct answer should have been True.

Would you like further details or clarification on the solution?

Here are 5 related questions to expand on this topic:

1. What is the geometric interpretation of linear independence for two vectors?
2. How can the concept of linear independence be extended to more than two vectors?
3. What is the difference between linear dependence and linear independence?
4. How does the determinant relate to checking linear independence for a system of vectors?
5. Can vectors in $\mathbb{R}^3$ ever be linearly dependent if they are non-parallel?

Tip: When checking for linear independence with two vectors, if one vector is a scalar multiple of the other, they are linearly dependent.