If we have a problem like problem 7 in section 5.3 (p. 373), is it sufficient to find a counter example to prove that W is not a subspace? Also, how can we prove W is a subspace if we can’t find any counter examples? I know to use theorem 2, but I don’t know what a sufficient explanation for each of the three properties of theorem 2 is sufficient. Thanks!
To prove $W$ is not a subspace, finding a counter example to any of (S1), (S2), (S3) in Theorem 2 (p.369)is sufficient.
To prove $W$ is a subspace, you need to verify all the subspace criteria in Theorem 2. Check out the following problems and solutions. These should illustrate how to write a solution.
- Subspaces of the vector space of all real valued function on the interval
- Determine whether a set of functions $f(x)$ such that $f(x)=f(1-x)$ is a subspace