Is $\{(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1)\}$ linearly INDEPENDENT in $\mathbb{R}^3$?
AYes
BOnly the first three
CNo (4 vectors in $\dim=3$ space)
DCannot tell without computing
Answer & Solution
Correct answer: C. No (4 vectors in $\dim=3$ space)
1. $\mathbb{R}^3$ has dimension 3.
2. THEOREM: any set of MORE than $n$ vectors in an $n$-dim space is linearly DEPENDENT.
3. So 4 vectors in $\mathbb{R}^3$ are AUTOMATICALLY dependent — option C.
4. Explicit dependence: $(1,1,1) = (1,0,0) + (0,1,0) + (0,0,1)$. The fourth vector is the sum of the first three.
5. Options A, B, D contradict the dimension argument.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §2.3 (Linear dependence in $\mathbb{R}^n$)._
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