Question

Determine whether the following are subspaces of c[−1, 1]:

A) The set of all constant functions f(x)=c, where c is a constant
B) The set of all functions g(x) such that g(0)=0
C) The set of all functions h(x) such that h(x)≥0 for all x in [−1,1]
D) The set of all functions p(x) such that p(1)=1

Answer 1

To determine if sets are subspaces of c[-1,1], we need to check closure under addition, scalar multiplication, and the presence of the zero vector. Sets A and B meet these conditions, while sets C and D do not.

Step-by-step explanation:

In order to determine whether a set is a subspace of c[-1,1], we need to check three conditions: closure under addition, closure under scalar multiplication, and the presence of the zero vector.

A) The set of all constant functions f(x)=c, where c is a constant, is a subspace of c[-1,1]. It is closed under addition and scalar multiplication, and the zero vector is present.

B) The set of all functions g(x) such that g(0)=0 is a subspace of c[-1,1]. It is also closed under addition and scalar multiplication, and the zero vector is present.

C) The set of all functions h(x) such that h(x)≥0 for all x in [-1,1] is not a subspace of c[-1,1] because it is not closed under scalar multiplication.

D) The set of all functions p(x) such that p(1)=1 is not a subspace of c[-1,1] because it is not closed under addition.