Exercise · ZW-VECTOR-SUBSPACE-010 · ★★★ Advanced
If \(W\) is a subspace of \(V\) and \(W\) contains a vector \(v\ne 0\), how many elements does \(W\) have?
Choose one option.
Show answer
- B. Infinitely many, because it contains every multiple \(cv\)
Explanation
By closure under scalars, \(cv\in W\) for every real \(c\), and these multiples are distinct when \(v\ne 0\). A subspace is either \(\{0\}\) or infinite.
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-002 A subset W of a vector space V is a subspace if and only if it…
- 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?