Determine which of the sets of vectors is linearly independent.
A: The set {p1, p2,p3} where p1(t) = 1, p2(t) = t^2, p3(t) = 3 + 4t
B: The set {p1, p2,p3} where p1(t) = t, p2(t) = t^2, p3(t) = 3t + 4t^2
C: The set {p1, p2,p3} where p1(t) = 1, p2(t) = t^2, p3(t) = 3 + 4t + t^2
Clearly A and C.
To determine which set of vectors is linearly independent, we need to check if any vector in the set can be written as a linear combination of the other vectors in the set.
For the set A: {p1, p2, p3} where p1(t) = 1, p2(t) = t^2, and p3(t) = 3 + 4t, we can see that none of the vectors can be written as a linear combination of the others. This means that the set A is linearly independent.
For the set B: {p1, p2, p3} where p1(t) = t, p2(t) = t^2, and p3(t) = 3t + 4t^2, we can see that p3(t) can be written as a linear combination of p2(t) and p1(t). We can express p3(t) as p3(t) = 4p2(t) + 3p1(t). This means that the set B is linearly dependent.
For the set C: {p1, p2, p3} where p1(t) = 1, p2(t) = t^2, and p3(t) = 3 + 4t + t^2, we can see that none of the vectors can be written as a linear combination of the others. This means that the set C is linearly independent.
Therefore, the sets A and C are linearly independent.