Vector group a1=(1, 1, 1), a2=(0,2.5), a3=(1.3,6), which of the following options is correct?

Question

Options

• ALinear correlation
• BLinear independence
• CaHa2+a3=0
• D2a1+a2+a3=0

• A
  74% (26)
• B
  3% (1)
• C
  6% (2)
• D
  17% (6)

Explanation

Interpreting the vectors as a1=(1,1,1), a2=(0,2,5), and a3=(1,3,6), these three vectors are linearly dependent because a1 + a2 - a3 = 0: adding a1 and a2 component-wise gives (1,3,6), which equals a3 exactly, so one vector is expressible as a combination of the others. Option B is wrong precisely because this dependency exists - a set is linearly independent only when no non-trivial combination equals zero. Option C fails because a1 + a2 + a3 = (2,6,12), not the zero vector, so the signs are off. Option D also fails: 2a1 + a2 + a3 = (3,7,13), which is nowhere near zero.

Memory tip: To quickly test linear dependence, try writing each vector as a sum of the others before reaching for row reduction - here, a3 = a1 + a2 is visible almost by inspection, which immediately rules out independence and exposes which of the combination formulas (C or D) is actually true.

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