Linear algebra MOC

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. linalg

∀[⃗𝐯∈𝐴]⃗𝐯∈span⁡(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

𝑘0⃗𝐯1+𝑘2⃗𝐯2+⋯+𝑘𝑛⃗𝐯𝑛=⃗𝟎

where a trivial solution is one where 𝑘1 =𝑘2 =⋯ =𝑘3 =0. This amounts to solving the homogenous system of linear equations

[⃗𝐯1⃗𝐯2⋯⃗𝐯3]⎡⎢ ⎢ ⎢ ⎢⎣𝑘1𝑘2⋮𝑘2⎤⎥ ⎥ ⎥ ⎥⎦=⃗𝟎

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

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Footnotes

1. 2022. Mathematical physics lecture notes, p. 136 ↩