Why is Linear Independence Important? Insights and Examples

[vigintitres]

Simply curious as to why it is so important to have linear independence? Is it solely based on the fact that linearly dependent vectors are simply multiples of one another and you receive no insight from their dependence relationship? I understand that LI vectors span a space and are necessary for a basis and all that but what are some examples of real systems that require LI vectors (or equations I suppose)?

I am almost done with my first semester of LA and I'm a mechanical engineer who is also a proposed controls-type engineer so this stuff is really cool to me. Thanks

 

[wisvuze]

To me, linear independent vectors are important to note because it is a way to "generalize" a vector space. If you can write other vectors from the same vector space as a linear combination of these linear independent vectors, then it is sort of "wasteful" or "not worth noting". This way, you can generalize vector spaces as the span of the maximal set of linearly independent vectors.

 

[Mark44]

Is it solely based on the fact that linearly dependent vectors are simply multiples of one another and you receive no insight from their dependence relationship?

Linearly dependent vectors are not just multiples of one another. A set of vectors can be linearly dependent if one of them is a linear combination of the others; that is, one of them is the sum of constant multiples of the others. For example, {(1, 0, 0), (0, 1, 0), (2, 3, 0)} is a linearly dependent set for which no vector is a multiple of any other, but the third vector in this list is a linear combination of the first two, with (2, 3, 0) = 2(1, 0, 0) + 3(0, 1, 0).

If you have a set of n linearly independent vectors in a space of dimension n, the set is a basis for the vector space. Every vector in the space can be written as a linear combination of the basis vectors.

 

[HallsofIvy]

Given a set of linearly dependent vectors, at least one of them can be written as a linear combination of the others and so is redundant. You can throw it out and do whatever you need to do with the smaller set of vectors.