Exercise · ZW-BASIS-AND-DIMENSION-009 · ★★★ Advanced
If \(\dim V=n\) and \(S\) is a linearly independent set of \(n\) vectors of \(V\), then:
Choose one option.
Show answer
- A. \(S\) is a basis of \(V\)
Explanation
An independent set with as many vectors as the dimension already spans \(V\); otherwise it could be extended to a basis with more than \(n\) vectors.
Review the concept: Basis and dimension
A basis \(\mathcal B\) of a vector space \(V\) is a linearly independent set that spans it; its number of vectors is the dimension of \(V\).
\[\mathcal{B}=\{v_1,\dots,v_n\},\quad \dim V=n\]
More practice problems: basis and dimension
- ZW-BASIS-AND-DIMENSION-001 What is the dimension of ℝ^4?
- ZW-BASIS-AND-DIMENSION-002 Which of these sets is a basis of ℝ^2?
- ZW-BASIS-AND-DIMENSION-003 How many vectors does any basis of ℝ^3 have?
- ZW-BASIS-AND-DIMENSION-004 Write (5,1)=a(1,1)+b(1,-1). What are a and b?
- ZW-BASIS-AND-DIMENSION-005 What is the dimension of the plane x+y+z=0 in ℝ^3?
- ZW-BASIS-AND-DIMENSION-006 What is the dimension of the span of (1,2,3), (2,4,6) and (0,1,0)?
- ZW-BASIS-AND-DIMENSION-007 What is the dimension of the space of polynomials of degree at most 3?
- ZW-BASIS-AND-DIMENSION-008 What is the dimension of the space of 2× 3 matrices?