Why are \(v_1\) & \(v_2\)
Linearly Dependent
in \(\mathbb{R}^3\)?
A complete walkthrough — definition, step-by-step proof, geometric interpretation, and key takeaways for the exam.
What is Linear Dependence?
Before solving, recall the core definition. This is what examiners expect you to state before your working.
A set of vectors \(\{v_1, v_2, \ldots, v_k\}\) in \(\mathbb{R}^n\) is linearly dependent if there exist scalars \(c_1, c_2, \ldots, c_k\), not all zero, such that $$c_1\,v_1 + c_2\,v_2 + \cdots + c_k\,v_k = \mathbf{0}.$$ For two vectors, this simplifies to: one is a scalar multiple of the other.
The key phrase is "not all zero." If every constant must equal zero, the vectors are linearly independent.
Proving Linear Dependence
Question: Explain why the following vectors are linearly dependent.
Solution:
Divide each component of \(v_2\) by the corresponding component of \(v_1\):
The ratio is constant = −5, confirming \(v_2\) is a scalar multiple of \(v_1\).
Rearranging \(v_2 = -5\,v_1\) gives:
Here \(c_1 = 5\) and \(c_2 = 1\) are not both zero — the definition is satisfied. ✓
What Does This Look Like?
Linearly dependent vectors in \(\mathbb{R}^3\) are collinear — they lie along the same line through the origin. No matter how many such vectors you collect, they can never span beyond that line. Their combined span contributes only 1 dimension, not 2.
Two linearly independent vectors span a full plane (a 2-D subspace). You need three linearly independent vectors to span all of \(\mathbb{R}^3\).
Key Takeaways
Three equivalent confirmations that \(\{v_1, v_2\}\) is linearly dependent:
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