MATH 01210 · Vector spaces

Linear independence / spanning sets

Definition

Vectors are linearly independent if c₁v₁+…+cₖvₖ=0 only when all cᵢ=0. They span a set S if every vector in S is a linear combination of them.

Independence fails iff one vector is a combination of others (or the matrix with those columns has nontrivial nullspace).

Spanning set can be redundant; basis = independent spanning set.

In Rⁿ, more than n vectors are always dependent.

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