Final answer:
To determine if the set of vectors is dependent or independent, we set up a matrix and look for the value of 'a' that allows for a non-trivial solution to the equation. For the vectors to be independent, no such value should exist. The condition for dependence or independence is found through solving the system.
Step-by-step explanation:
To determine whether the set of vectors provided is dependent or independent, one can perform operations such as setting up a matrix with these vectors and trying to reduce it to its echelon form or use methods such as the determinant to identify dependencies. Specifically, we are examining vectors in the form:
[5][-10][5][-1]
[4][a][2][-3]
[3][2][-2][4]
For these vectors to be dependent, there must exist a non-trivial solution to the equation a1v1 + a2v2 + a3v3 + a4v4 = 0, where a1, a2, a3, and a4 are scalars, and v1 through v4 are our vectors. If there is a specific value of 'a' that allows for a non-trivial solution, that value will make the set dependent. If no such value exists, the set is independent.
For the given vectors, we look for a condition where one of the vectors can be expressed as a linear combination of the others. If 'a' permits such an expression, it leads to dependence. Conversely, if no such value exists, the vectors would be independent regardless of the value of 'a'.
In conclusion, the value(s) of 'a' that lead to dependence or independence can be identified by setting up the appropriate system and solving for the conditions where a non-trivial solution exists.