(a) Let S be the set of solutions y(t) to the differential equation dy/dt = -y for t> 0. With addition and scalar multiplication of elements defined in the usual way, is S a vector space?
(b)Let T be the set of solution y(t) to the differential equation dy/dt = 1 - y for t > 0. With addition and scalar multiplication of elements defined in the usual way, is T a vector space?
To determine whether the set S is a vector space, we need to check if it satisfies all the axioms of a vector space.
(a) The set S consists of solutions y(t) to the differential equation dy/dt = -y for t > 0.
To verify if S is a vector space, we need to check the following axioms:
1. Closure under addition: For any two solutions y1(t) and y2(t) in S, their sum y1(t) + y2(t) should also be a solution. Let's check if dy1/dt = -y1 and dy2/dt = -y2, then dy1(t)/dt + dy2(t)/dt = -y1 - y2 = -(y1 + y2). Therefore, the sum of two solutions is also a solution.
2. Closure under scalar multiplication: For any scalar c and a solution y(t) in S, cy(t) should also be a solution. We know that dy(t)/dt = -y. Let's check if d(cy(t))/dt = -cy(t). Since d(cy(t))/dt = c(dy(t)/dt) = -cy(t), the scalar multiplication of a solution is also a solution.
3. Commutativity of addition: For any two solutions y1(t) and y2(t) in S, y1(t) + y2(t) = y2(t) + y1(t). This property holds because addition of functions is commutative.
4. Associativity of addition: For any three solutions y1(t), y2(t), and y3(t) in S, (y1(t) + y2(t)) + y3(t) = y1(t) + (y2(t) + y3(t)). This property holds because addition of functions is associative.
5. Identity element of addition: There should exist an identity element, denoted as 0, such that for any solution y(t) in S, y(t) + 0 = y(t). The zero function, where every value is zero, satisfies this property since y(t) + 0 = y(t) + 0 = y(t).
6. Inverse element of addition: For any solution y(t) in S, there should be an element -y(t) in S such that y(t) + (-y(t)) = 0. In this case, if y(t) is a solution, then -y(t) is also a solution since dy(t)/dt = -y(t) implies d(-y(t))/dt = -(dy(t)/dt) = -(-y(t)) = y(t).
With all these axioms satisfied, we can conclude that S is a vector space.
(b) Similarly, for the set T, consisting of solutions y(t) to the differential equation dy/dt = 1 - y for t > 0, we can follow the same process to determine if it is a vector space.
1. Closure under addition: For any two solutions y1(t) and y2(t) in T, their sum y1(t) + y2(t) should also be a solution. Let's check if dy1/dt = 1-y1 and dy2/dt = 1-y2, then dy1(t)/dt + dy2(t)/dt = (1-y1) + (1-y2) = 2 - (y1 + y2). Therefore, the sum of two solutions is also a solution.
2. Closure under scalar multiplication: For any scalar c and a solution y(t) in T, cy(t) should also be a solution. We know that dy(t)/dt = 1 - y. Let's check if d(cy(t))/dt = 1 - cy(t). Since d(cy(t))/dt = c(dy(t)/dt) = 1 - cy(t), the scalar multiplication of a solution is also a solution.
3. Commutativity of addition: Holds, as addition of functions is commutative.
4. Associativity of addition: Holds, as addition of functions is associative.
5. Identity element of addition: Holds, as the zero function satisfies y(t) + 0 = y(t).
6. Inverse element of addition: Holds, as -y(t) is also a solution, satisfying y(t) + (-y(t)) = 0.
Based on these axioms, we can conclude that T is also a vector space.