Questions: Linear Independence and Linear Dependence

Question 1

You have three vectors in ℝ² (the 2D plane). Can they be linearly independent?

• A. Yes, as long as no two of the three vectors are parallel to each other
• B. Yes, provided all three vectors are nonzero
• C. No — any set of three vectors in ℝ² must be linearly dependent
• D. Only if all three vectors are mutually orthogonal

Answer: (C)

ℝ² is 2-dimensional, meaning at most 2 linearly independent vectors can exist in it. A third vector must lie in the plane spanned by the first two — it is always expressible as a linear combination of them. This is a fundamental constraint: you cannot have more linearly independent vectors than the dimension of the space. The options about 'not parallel' or 'nonzero' represent common misconceptions — independence is about dimensional redundancy, not just pairwise geometric distinctness.

Question 2

For vectors v₁, v₂, v₃, suppose the equation c₁v₁ + c₂v₂ + c₃v₃ = 0 has the solution c₁ = 2, c₂ = −1, c₃ = 1. What can you conclude?

• A. The vectors are linearly independent, because the coefficients are not all equal
• B. The vectors are linearly dependent, because a nontrivial combination equals zero
• C. We need to know whether the vectors are in ℝ² or ℝ³ before concluding anything
• D. The vectors are orthogonal, since combining them in this way yields zero

Answer: (B)

Linear independence requires that the only solution to c₁v₁ + c₂v₂ + c₃v₃ = 0 is the trivial solution (all cᵢ = 0). Here we have a nontrivial solution — at least one coefficient is nonzero (c₁ = 2). This immediately establishes linear dependence. From c₁ = 2 ≠ 0, we can solve: v₁ = (1/2)v₂ − (1/2)v₃, showing v₁ is redundant. The dimension of the space is irrelevant to this conclusion; what matters is whether a nontrivial combination can be zero.

Question 3

The zero vector cannot be a member of a linearly independent set, even if all other vectors in the set are nonzero.

• T. True
• F. False

Answer: True

If the zero vector 0 is in a set {0, v₂, ..., vₖ}, then the equation 1·0 + 0·v₂ + ··· + 0·vₖ = 0 is a nontrivial solution (the coefficient of 0 is 1, not 0). This violates the definition of linear independence, which requires that the only solution be all-zero coefficients. The zero vector is always redundant: you can 'zero it out' with a nonzero coefficient while leaving the rest unchanged.

Question 4

If a set of vectors is linearly dependent, then at least one of them can be written as a linear combination of the others.

• T. True
• F. False

Answer: True

This is exactly what dependence means in practice. If there is a nontrivial combination c₁v₁ + ··· + cₖvₖ = 0 with some cᵢ ≠ 0, then vᵢ = −(c₁/cᵢ)v₁ − ··· (omitting vᵢ) — it is reachable from the others by scaling and adding. Linear independence means no vector can be reached this way; linear dependence means at least one can. This is the geometric content of the algebraic definition: dependence equals redundancy.

Question 5

Why does the definition of linear independence use the algebraic condition 'c₁v₁ + ⋯ + cₖvₖ = 0 implies all cᵢ = 0' rather than a simpler geometric condition like 'no two vectors point in the same direction'?

Model answer: The geometric condition 'no two vectors are parallel' only catches pairwise collinearity — it misses the case where three or more vectors are dependent even though no two of them are parallel. For example, in ℝ², the vectors (1,0), (0,1), and (1,1) have no two pairs that are parallel, yet they are linearly dependent because (1,1) = 1·(1,0) + 1·(0,1). The algebraic definition captures all forms of redundancy simultaneously: any vector expressible as a combination of the others will produce a nontrivial solution to the zero-combination equation. The algebraic condition is both necessary and sufficient; the pairwise geometric condition is neither.

Geometric intuition about 'direction' breaks down in higher dimensions and for more than two vectors. The algebraic definition is universal — it applies in any vector space of any dimension, including abstract ones where 'direction' has no geometric meaning.