The vectors [3], [0], and [5] are linearly independent.
To determine if the vectors are linearly independent, we can set up an equation of linear dependence and check if the only solution is the trivial solution (where all scalars are zero).
Let's assume that there exist scalars a, b, and c (not all zero) such that the equation below is true:
a[3] + b[0] + c[5] = [0].
Simplifying this equation, we get:
[3a + 5c] = [0].
For this equation to hold true, we must have 3a + 5c = 0.
Since the equation 3a + 5c = 0 has only the trivial solution (a = 0, c = 0), we can conclude that the vectors [3], [0], and [5] are linearly independent.
In the given equation:
[-2] + [0], + [3] = [0]
[-5] [-5] [-3] [0]
There are no non-zero scalars that satisfy this equation. Therefore, the only solution that makes this equation true is a = b = c = 0, which corresponds to the trivial solution. This further confirms that the vectors [3], [0], and [5] are linearly independent.