Question

Difference between linear dependence and independence in linear algebra

Answer 1

It’s important to understand the concept of a linear combination. A linear combination of a set of vectors v1,...,vnv1,...,vn is a sum of the form a1v1+...+anvna1v1+...+anvn. The question is when it’s possible for a linear combination to be the zero vector. It is always possible to get the zero vector by setting a1=...=an=0a1=...=an=0. So set aside that possibility, and consider instead, when is it possible to get the zero vector some other way (using a1,...,ana1,...,an at least some of which are not zero). Linearly dependent means “yes, you can”, linearly independent means, “no, you can’t”.

So for example, a single vector v1v1 being linearly dependent means that you can multiply it by a non-zero scalar and get the zero vector. This is only possible if you started out with the zero vector.

If two vectors are linearly dependent, a1v1+a2v2=0a1v1+a2v2=0 where a1a1 and a2a2 are not both 0, it could be because one of the vectors is the zero vector. If not, then it must be that neither a1a1 nor a2a2 is zero, and we can write either v1=−(a2/a1)v2v1=−(a2/a1)v2 or v2=−(a1/a2)v1v2=−(a1/a2)v1, i.e. each is a multiple of the other. Or to put it another way, the two vectors are on a common line through the origin.

For three vectors to be linearly dependent means that they are on a plane through the origin.

In general, a finite set of vectors is linearly dependent when there is a vector subspace (which by definition includes the origin) which contains them, but has dimension lower than the number of vectors. The smallest vector subspace containing a set of vectors is called their span. So linearly independent means the opposite thing: that the span has dimension equal to the number of vectors.

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