Final answer:
To check if set S is linearly independent, solve the equation a(1, 4, 1) + b(2, 1, 3) + c(-2, 9, 2) = (0, 0, 0). To check if set S is a basis for R3, determine if the vectors are linearly independent and if they span R3.
Step-by-step explanation:
A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. To check if the set S = {(1, 4, 1), (2, 1, 3), (-2, 9, 2)} is linearly independent, we need to see if the equation a(1, 4, 1) + b(2, 1, 3) + c(-2, 9, 2) = (0, 0, 0) only has the trivial solution a = 0, b = 0, c = 0. Solving this system of equations, we can determine that the set S is linearly independent.
A set of vectors is a basis for R3 if it is linearly independent and spans R3. To check if the set S = {(0, 3, -2), (4, 0, 3), (-8, 15, -16)} is a basis for R3, we need to see if the vectors in S are linearly independent and if they span R3. Using the same method as before, we can determine that the set S is linearly independent. To check if they span R3, we need to see if any vector in R3 can be written as a linear combination of the vectors in S. By performing row operations on the augmented matrix, we can determine that the vectors in S span R3. Therefore, the set S = {(0, 3, -2), (4, 0, 3), (-8, 15, -16)} is a basis for R3.