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take the first four column and solve by determinant then find the value of a 

The determinant of the \( 4 \times 4 \) matrix formed by the first four columns is:

\[
\text{det} = a^5 - 4a^3 - 17a^2 + 7a + 14
\]

For the set of vectors to be linearly dependent, the determinant must be equal to zero. Therefore, we need to solve:

\[
a^5 - 4a^3 - 17a^2 + 7a + 14 = 0
\]

Would you like me to solve this equation for the values of \( a \)? 

Here are some related questions to think about:

1. How do we know if a set of vectors is linearly independent or dependent?
2. What is the geometric interpretation of linear dependence?
3. What happens when the determinant of a matrix is zero?
4. How does the value of \(a\) affect the matrix's linear dependence?
5. What are some other methods for checking linear dependence besides using determinants?

**Tip**: When solving higher-degree polynomial equations, factoring or numerical methods are often used to find the roots.

For what value(s) of a is the set of vectors S linearly dependent?

Explore how to determine the values of 'a' for which a set of vectors becomes linearly dependent. The solution involves calculating the determinant of a 4x4 matrix and solving a polynomial equation.

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