I want to make sure that I'm correct with my answers to whether these three sets are vector spaces or not:
- A = $\{(a,b,c,d) \in \mathbb{R}^4: a +5c +d = 0\}$
- B = $\{ f \in C[0,1]: f(0) =1\}$
- C = $\{ f \in C[-1,1]: \int_{-1}^1 f(x) \ dx = 0\}$
A is a vector space because it's a subspace to $\mathbb{R}^4$ (a vector space) and it satisfies that vector addition and scalar multiplication generates vectors inside of the same subspace. It's also given that the zero vector is part of the set.
B is not a vector space because the zero-function is not an element in the set.
C is a vector space because $F(x)=0$ on the interval [-1,1] implies that $f(x)=0$ somewhere on that interval. Continuous functions have well defined function addition and scalar multiplication so we know that it satisfies the conditions for a subspace of a vector space.
Best Answer
Yes, very good except that the argument in C needs a little attention. When you say "f(x)=0 somewhere on that interval" I assume you wanted to say "the zero function has zero integral". There is of course a world of difference between being zero at one point and being identically zero on an interval.