Vectors & vector spaces
The idea
Linear algebra begins by abstracting what you already do with arrows in the plane: add them, scale them. A vector space is any collection closed under those two operations — column vectors in Rⁿ, but equally polynomials, matrices, or solutions of a linear differential equation. The central vocabulary is span (everything reachable by linear combinations), linear independence (no vector is redundant), basis (an independent spanning set), and dimension (the size of any basis).
The productive mental model is reachability and redundancy. The span of a set is the territory its combinations cover; independence asks whether each vector adds new territory. Three vectors in R³ might span all of space, or only a plane, or a line — counting vectors tells you nothing until you test independence, usually by asking whether some combination equals the zero vector with not-all-zero coefficients.
The misconception to flush out: independence is not pairwise. Three vectors can be pairwise non-parallel yet still dependent, because one may lie in the plane spanned by the other two. Dependence is a team property — it is about whether the whole set carries redundancy, not whether any two members look alike.
Worked example
Determine whether the vectors v₁ = (1, 2, 3), v₂ = (2, 1, 0), and v₃ = (4, 5, 6) are linearly independent in R³, and describe their span.
- Test whether v₃ is reachable from the first two: seek scalars a and b with a v₁ + b v₂ = v₃, which gives the system a + 2b = 4, 2a + b = 5, 3a = 6.
- The third equation involves only a (since v₂ has third coordinate 0): a = 2. Substitute into the first: 2 + 2b = 4, so b = 1.
- Verify against the unused second equation: 2(2) + 1 = 5, which matches — the system is consistent, so v₃ = 2v₁ + v₂ exactly.
- Conclude dependence: v₃ adds no new territory, so the set is linearly dependent. Note that no two of these vectors are parallel — pairwise checks would have missed this completely.
- Describe the span: v₁ and v₂ are independent (neither is a scalar multiple of the other), so the three vectors together span only the plane through the origin containing v₁ and v₂ — a 2-dimensional subspace of R³, with {v₁, v₂} as a basis.
Answer. The set is linearly dependent, since v₃ = 2v₁ + v₂; the span is the 2-dimensional plane spanned by v₁ and v₂.
Check your understanding
- Why can three pairwise non-parallel vectors in R³ still fail to be independent?
- How does the definition of independence via the zero vector connect to the idea of no vector being redundant?
- What does it tell you geometrically when n vectors in Rⁿ fail to span the whole space?
- Why do sets like polynomials of degree at most 3 deserve the name vector space, and what plays the role of the arrows?
Build the foundations first
Vectors & vector spaces builds on these concepts. If any feel shaky, start there.