Final answer:
The set of vectors S = {x^2, 5 + x^2} in P_2 is linearly independent because the only solution to their linear combination equaling the zero vector is the trivial solution where all coefficients are zero.
Step-by-step explanation:
To determine whether the set of vectors S = {x^2, 5 + x^2} in P_2 is linearly independent or linearly dependent, we can set up a linear combination of these vectors equal to the zero vector and solve for the coefficients. If the only solution for the coefficients is the trivial solution (where all coefficients are zero), then the set of vectors is linearly independent. If there are non-trivial solutions (where at least one coefficient is not zero), then the set is linearly dependent.
We start by setting up the equation ax^2 + b(5 + x^2) = 0. Expanding and grouping like terms, we get (a + b)x^2 + 5b = 0. This equation must hold for all values of x, so both coefficients of terms must be zero: a + b = 0 and 5b = 0. Solving this system, we find that b = 0 and thus a = 0. Since the only solution is the trivial solution, the vectors are linearly independent.