Exercise · ZW-LINEAR-INDEPENDENCE-009 · ★★★ Advanced
If \(u\) and \(v\) are linearly independent, which of these sets is dependent?
Choose one option.
Show answer
- B. \(\{u-v,\ v-u\}\)
Explanation
\(v-u=-(u-v)\), so \((u-v)+(v-u)=0\). In the other sets, equating to \(0\) forces zero coefficients.
Review the concept: Linear independence
A set of vectors is linearly independent if the only linear combination of them that gives the zero vector is the one with all coefficients equal to zero.
\[c_1v_1+\cdots+c_kv_k=0\ \Rightarrow\ c_1=\cdots=c_k=0\]
More practice problems: linear independence
- ZW-LINEAR-INDEPENDENCE-001 Are (1,2) and (2,4) linearly independent?
- ZW-LINEAR-INDEPENDENCE-002 Which of these subsets of ℝ^2 is linearly independent?
- ZW-LINEAR-INDEPENDENCE-003 A set containing the zero vector is always:
- ZW-LINEAR-INDEPENDENCE-004 For v_1=(1,2,3) and v_2=(2,4,6), which coefficients (c_1,c_2), not both zero, give c_1v_1+c_2v_2=0?
- ZW-LINEAR-INDEPENDENCE-005 For which value of k are (1,2) and (3,k) linearly dependent?
- ZW-LINEAR-INDEPENDENCE-006 The vectors (1,0,1), (0,1,1) and (1,1,2) are:
- ZW-LINEAR-INDEPENDENCE-007 A set of 4 vectors in ℝ^3 is always:
- ZW-LINEAR-INDEPENDENCE-008 For which value of k are (1,0,2), (0,1,1) and (1,1,k) linearly dependent?