In Exercises 28-33, determine conditions on the scalars so that the set of vectors is linearly dependent. 28. $\mathbf{v}_{1}=\left[\begin{array}{l}1 \\ a\end{array}\right], \quad \mathbf{v}_{2}=\left[\begin{array}{l}2 \\ 3\end{array}\right]$

Step 1 ：A set of vectors is linearly dependent if there exist scalars, not all zero, such that a linear combination of the vectors equals the zero vector. In this case, we need to find conditions on the scalar 'a' such that the vectors v1 and v2 are linearly dependent. This means we need to solve the following equation for scalars c1 and c2: \(c1 * v1 + c2 * v2 = 0\)

Step 2 ：This gives us a system of two equations: \(c1 + 2*c2 = 0\) and \(a*c1 + 3*c2 = 0\)

Step 3 ：We can solve this system of equations to find conditions on 'a' for which the system has a nontrivial solution (i.e., a solution other than c1 = c2 = 0).

Step 4 ：The solution to the system of equations is c1 = c2 = 0. This is the trivial solution, which means that the vectors are linearly independent for all values of 'a'. However, we are looking for conditions on 'a' that make the vectors linearly dependent, which means we need a nontrivial solution.

Step 5 ：A nontrivial solution exists when the determinant of the coefficients of the system of equations is zero. So, we need to find the value of 'a' that makes the determinant zero.

Step 6 ：The determinant of the matrix of coefficients is \(3 - 2*a\). Setting this equal to zero gives us the condition on 'a' for which the vectors are linearly dependent.

Step 7 ：Final Answer: The set of vectors is linearly dependent when \(a = \boxed{\frac{3}{2}}\).