Final answer:
To determine if the set {r, u, x} is linearly independent, we substitute the value of x into a linear combination of r, u, and x, and check if the coefficients are all zero. If the coefficients are all zero, the set is linearly independent.
Step-by-step explanation:
To determine whether the set t = {r, u, x} is linearly independent, we need to check if the linear combination of these vectors equals zero only when all the coefficients are zero.
If we assume that the linear combination c1r + c2u + c3x = 0, where c1, c2, and c3 are constants, we can substitute the value of x into the equation: c1r + c2u + c3(3r + 4u + d) = 0.
As s = {r, u, d} is a set of linearly independent vectors, r, u, and d are linearly independent. Therefore, we can rewrite the equation as (c1 + 3c3)r + (c2 + 4c3)u + c3d = 0.
In order for this equation to hold true, all the coefficients must equal zero. This means c1 + 3c3 = 0, c2 + 4c3 = 0, and c3 = 0. Solving these equations, we find that c1 = c2 = c3 = 0. Since the coefficients are all zero, the set t = {r, u, x} is linearly independent.