Subspace Quiz

Subspaces

• A subspace is a nonempty subset W of a vector space V that is also a vector space with the same operations of addition and scalar multiplication.
• A subspace W of a vector space V must satisfy the following conditions:
  • W is nonempty
  • W is closed under addition
  • W is closed under scalar multiplication
  • W satisfies the ten axioms of a vector space

Example 1: W = {(x, y) | x + 2y = 1}

• This is not a subspace of R² because it does not contain the zero vector (0, 0).
• Consider v = (1, 0) ∈ W. (-1)v = (-1, 0) ∉ W, so W is not closed under scalar multiplication.

Example 2: W = {(x₁, x₂) | x₁ ≥ 0 and x₂ ≥ 0}

• This is not a subspace of R² because it is not closed under scalar multiplication.
• Let u = (1, 1) ∈ W. (-1)u = (-1, -1) ∉ W.

Trivial Subspaces

• Every vector space V has at least two subspaces: the zero vector space {0} and V itself.
• These two subspaces are called trivial subspaces.
• Any subspaces other than these two are called proper (or nontrivial) subspaces.