Linear (in)dependence

Essentially, a set of vectors $𝐴$ is linearly dependent iff at least one of its contained vectors can be derived from a linear combination of the others. #m/def/linalg

\forall [⃗ 𝐯 \in 𝐴] ⃗ 𝐯 \in s p a n (𝐴 ∖ {⃗ 𝐯})

If the inverse is true, the vectors are linearly independent. Such a set it said to be an efficient spanning set, since none of the set's members are redundant, i.e. removing any vector from the set would change the span. A spanning set that is linearly independent forms a basis for its span.

An infinite set of vectors is linearly independent iff. every finite subset is linearly independent.¹

Proving linear independence

Proving a set of vectors $𝐴$ is linearly independent amounts to showing that there are no non-trivial solutions to the equation

𝑘 0 ⃗ 𝐯 1 + 𝑘 2 ⃗ 𝐯 2 + \dots + 𝑘 𝑛 ⃗ 𝐯 𝑛 = ⃗ 𝟎

where a trivial solution is one where $𝑘 1 = 𝑘 2 = \dots = 𝑘 3 = 0$ . This amounts to solving the homogenous system of linear equations

[⃗ 𝐯 1 ⃗ 𝐯 2 \dots ⃗ 𝐯 3] ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑘 1 𝑘 2 ⋮ 𝑘 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⃗ 𝟎

where a single, trivial solution indicates the set is indeed independent — otherwise, infinite solutions will be given.

#state/tidy | #SemBr

2022. Mathematical physics lecture notes, p. 136 ↩

Linear (in)dependence

Proving linear independence

Footnotes