Practice question · Fill in the blanks
Complete the consequence of closure under scalar multiplication.
Because a nonempty subspace is closed under scaling by every scalar, taking c = 0 shows that it must ______.
Word bank: be infinite · contain the zero vector · contain a basis · be closed under addition
Hints
- Apply the closure rule with the least interesting scalar available.
- 0 times any vector is the zero vector, and closure says the result stays inside.
Show the answer
Because a nonempty subspace is closed under scaling by every scalar, taking c = 0 shows that it must contain the zero vector.
Why
Pick any v in W; closure gives 0v = 0 in W. This is why the zero-vector test is legitimate rather than an extra axiom, it is a consequence of closure, and it rejects every set offset from the origin in one line.
Practise Subspaces
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