Final answer:
To find values making the third vector linearly independent, choose values for a, b, c, d, e not linear combinations of correspondents in given vectors. For certainty, perform matrix determinant or row reduction for full rank confirmation.
Step-by-step explanation:
To determine the values of a, b, c, d, and e that make the vectors linearly independent, we need to ensure that the third vector is not a linear combination of the first two vectors. This means no scalar multiples of the first two vectors can add up to the third vector. The first two vectors are [4, 9, -3, -5, 0] and [5, 4, 4, -4, 3]. Let's represent the incomplete vector as [a, b, c, d, e].
One strategy to find a set of values for the third vector that guarantees linear independence is to pick a value for a that is not a linear combination of the first components of the existing vectors (4 and 5), perhaps a = 1. Subsequently, we can pick a value for b that is distinct from any combination of 9 and 4, like b = 1. We follow a similar strategy for the rest of the components, leading us to a possibly independent vector, for instance, [1, 1, 1, 1, 1].
However, to conclusively ensure that our chosen vector is linearly independent from the given vectors, we should verify by forming a matrix with the vectors as rows and calculating its determinant or by using row-reduction techniques to confirm that the matrix is of full rank (rank 3 for three vectors in five dimensions), which means that the vectors are linearly independent.