Linear algebra Consider the vector space V consisting of all

Linear algebra

Consider the vector space V consisting of all differentiable functions from R to R. The subset W_0 V consists of all functions in V for which f\'(3) = 0, and W_1 V consists of all functions in V for which f\'(3) = 1. Prove that W_0 is and W_1 is not a linear subspace of V.

Solution

First we check W0

1. 0 belongs to W0 ie the 0 function which sends each x to 0

2. Let, f and g be in W0

(f+g)\'(3)=f\'(3)+g\'(3)=0+0=0

f+g is in W0

3. Let, c be a scalar and f be in W0

(cf)\'(3)=cf\'(3)=0

Hence, cf is in W0

Hence, W0 is a linear subspace of V

Now we check W1

1. 0 does nto belong to W1 as derivative of 0 function is also the 0 function.

Hence, W1 is not a linear subspace of V