Exercise · ZW-VECTOR-SUBSPACE-002 · ★☆☆ Basic
A subset \(W\) of a vector space \(V\) is a subspace if and only if it…
Choose one option.
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- C. is nonempty and closed under addition and under scalar multiplication
Explanation
It is enough to check three things: \(W\) is nonempty (equivalently contains zero), closure under addition and closure under scalar multiplication; the remaining axioms are inherited from \(V\).
Review the concept: Vector subspace
A vector subspace \(W\) of a vector space \(V\) is a nonempty subset that is itself a vector space under the same operations.
\[0\in W,\quad u,v\in W\Rightarrow u+v\in W,\quad c\in\mathbb{R},\ v\in W\Rightarrow cv\in W\]
More practice problems: vector subspace
- ZW-VECTOR-SUBSPACE-001 Which of these sets is a subspace of ℝ^2?
- ZW-VECTOR-SUBSPACE-003 Let W=\(x,y,z): x+y+z=0\. Which of these vectors belongs to W?
- ZW-VECTOR-SUBSPACE-004 Why is \(x,y): x+y=1\ not a subspace of ℝ^2?
- ZW-VECTOR-SUBSPACE-005 Which of these subsets of ℝ^3 is NOT a subspace?
- ZW-VECTOR-SUBSPACE-006 Inside the space of functions f:ℝ→ℝ, which set is a subspace?
- ZW-VECTOR-SUBSPACE-007 Let U=\(x,y,z): x=2y\ and W=\(x,y,z): z=0\. What do the vectors of U∩ W look like?
- ZW-VECTOR-SUBSPACE-008 In the space of polynomials of degree at most 2, which set is a subspace?
- ZW-VECTOR-SUBSPACE-009 Let U be the x-axis and W the y-axis in ℝ^2. Why is U∪ W not a subspace?