Physics I / Vector Spaces and Subspaces
Practice question · Put in order

Order these steps to check if a set S is a subspace of ℝⁿ.

Hints
  1. The zero vector must be present before closure is worth checking.
  2. Closure has two separate parts: under addition and under scaling.
Show the answer
  1. Verify the zero vector is in S
  2. Check closure under addition: u,v ∈S\in S ⟹ u+v ∈S\in S
  3. Check closure under scalar multiplication: u ∈S,c\in S, c ∈ ℝ ⟹ cu ∈S\in S
  4. Conclude S is a subspace
Why

All three conditions must hold: contains zero, closed under addition and scaling.

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