Practice question · Multiple choice
Every subspace must contain the zero vector. Why does that single requirement rule out so many otherwise reasonable-looking sets?
Hints
- Picture all the lines in the plane. What fraction pass through the origin?
- Ask how long it takes to check this compared with checking closure.
Show the answer
B. Because it excludes everything that misses the origin
Why
Almost every line and plane you can draw misses the origin, and one glance eliminates all of them. It is also not an independent axiom, scaling any element by 0 produces zero, so it is a fast necessary condition extracted from closure, which is exactly what makes it a good first filter.
Practise Subspaces
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