Question

Each subset W⊆Rⁿ is not a subspace. Which defining properties fail (circle one)? Find concrete examples demonstrating failure of each property. Is 0∈W or not (circle one)?

(a) W={(x,y)∈R² :x≥y}
(b) W={(x,y,z)∈R³:y=2x+1}
(c) W={(x,x,−2,w)∈R⁴:x,w∈R}

Answer 1

The subsets W given in (a) and (b) do not satisfy all three properties of a subspace as they don't contain the zero vector nor are closed under addition or scalar multiplication. Subset (c) satisfies all properties, containing the zero vector and closing under both operations, and is a subspace of R^4.

Step-by-step explanation:

The question asks for an analysis of subsets of R^n to see if they qualify as subspaces. A subspace must (1) contain the zero vector, (2) be closed under vector addition, and (3) be closed under scalar multiplication. Let's examine each set:

• (a) W={(x,y) ∈ R² : x ≥ y}: This set does not contain the zero vector (0, 0) since there's no guarantee x=y. Also, it is not closed under vector addition or scalar multiplication (negative scalars).
• (b) W={(x,y,z) ∈ R³ : y=2x+1}: This set does not contain the zero vector since y=1 when x=0. It fails both addition and scalar multiplication tests for subspace as well.
• (c) W={(x,x,−2,w) ∈ R´ : x,w ∈ R}: This set does contain the zero vector (0,0,0,0). It is closed under vector addition and scalar multiplication, making it a subspace.