To understand if vectors are linearly independent using determinants, we can look at an important property of matrices and how they relate to their vectors.

Vectors and Their Matrix:
Imagine we have $n$ vectors, which we call $v_1, v_2, \ldots, v_n$, in a space called $\mathbb{R}^m$. We can create a matrix, $A$, by listing these vectors as columns like this:

$$A = [v_1 \, v_2 \, \cdots \, v_n].$$

Square Matrix Requirement:
To use the determinant, we need one important rule. The number of dimensions $m$ should be at least as great as the number of vectors $n$. In easier terms, if we have more vectors than dimensions, they will depend on each other. Also, the vectors should be in $\mathbb{R}^n$ so that $A$ becomes a square matrix with $n$ rows and $n$ columns.

Determinant Check:
The next step is to calculate the determinant of this matrix. If we find that $\det(A) \neq 0$, it means the vectors are linearly independent. If we get $\det(A) = 0$, then the vectors are linearly dependent. A non-zero determinant shows that the matrix can be inverted, which only happens if no vector can be written as a mix of the others.

Applications:
This method is very helpful for understanding the reach of vector sets. For instance, if we have three vectors in $\mathbb{R}^3$ and calculate the determinant of their matrix, finding a non-zero result tells us these vectors fill up the three-dimensional space.

Geometric Interpretation:
Geometrically, a non-zero determinant means that the space formed by the vectors has volume. This shows that they form a basis for the space.

In conclusion, determinants are a powerful tool for figuring out if vectors are independent or dependent. By looking at the determinant of the matrix made from these vectors, we can easily see how they relate to each other in their space.