What does it mean for two vectors to be linearly independent?

Question

Options

• AOne vector can be written as a linear combination of the other.
• BThe vectors have unlimited span and can create new vectors in any direction.
• CThe vectors exist on the same line and have the same direction.
• DThe dot product of the vectors is 0.

• B
  92% (22)
• C
  4% (1)
• D
  4% (1)

Explanation

Note: Option B is the marked answer, but it's worth flagging that it's imprecisely worded - "any direction" overstates the case. The better framing is below.

Two linearly independent vectors point in genuinely different directions, meaning neither can be expressed as a scalar multiple (or linear combination) of the other - together they "reach" more of the space, which is the spirit of option B. Option A is actually the definition of linear dependence, the opposite concept: if one vector can be written as a combination of the other, they're dependent, not independent. Option C describes collinear (parallel) vectors, which are a textbook example of linear dependence since one is just a scalar multiple of the other. Option D confuses linear independence with orthogonality - for example, the vectors (1, 0) and (1, 1) are linearly independent yet have a nonzero dot product.

Memory tip: Think "independent = not trapped on the same line." Dependent vectors are redundant - one is just a stretched version of the other. Independent vectors give you genuine new directions to explore.

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