The best definition of Linearly Independence

Question

I was wondering about the definition of Linearly Independent sets of vectors of some textbooks and how it really depends on the author vision of it.

Usually we find the definition:

A set of vectors $A=\{\vec{v}_1,\vec{v}_2,...,\vec{v}_n\}$ is said to be linearly independent if and only if the equation $\alpha_1\vec{v}_1+\alpha_2\vec{v}_2+...+\alpha_n\vec{v}_n=0$ has only one solution (the trivial one: $\alpha_1=\alpha_2=...=\alpha_n=0$).

This is the definition we see, for instance, in wikipedia.

However, another sources point out for another definition:

A set of vectors $A=\{\vec{v}_1,\vec{v}_2,...,\vec{v}_n\}$ is said to be linearly independent if and only if none of the vectors $\vec{v}_1,\vec{v}_2,...,\vec{v}_n$ can be written as a linear combination of the others.

We can find this definition on cliffsnotes, among other sites.

I know that both definitions are equivalent (and when an author uses one of them as definition, concludes the other as theorem), and I know also that definitions in math are not as regularized as they should.

What I would like no know is if there is a more "correct" way than the other. What motivated the concept of Linearly Independent Set? Maybe an explanation of this reason helps me to decide how I must write my notes.

Thanks.

Answer 1

Just for the fun of it, let me add a third definition:

A collection of vectors $v_1, \dots, v_n$ is linearly independent if and only if the following holds: When $v$ is a linear combination of $v_1, \dots, v_n$, that is $$ v = \alpha_1 v_1 + \cdots + \alpha_n v_n, $$ then the coefficients $\alpha_1,\dots,\alpha_n$ are uniquely determined.

Each of the three descriptions has their advantages and disadvantages and I think when learning about linear independence one should learn all three of them and really understand how and why they are describing the same property (are equivalent).

A good way to introduce this topic in my opinion is with a theorem instead of a definition:

Theorem. For a collection of vectors $v_1,\dots,v_n$ the following conditions are equivalent:

1. …
2. …
3. …

If any one (and hence all) of the conditions is satisfied, we call the collection linearly independent.

Answer 2

Even though the two definitions are equivalent, I have often seen the first definition being used more often than the second. However, I think that the second definition is more useful since it provides an intuitive point of view for linear independence. Usually, people are first introduced to linear combinations and then to linear independence, and defining linear independence in terms of linear combinations provides a smoother transition. I should also add that I personally think that the second definition motivated linear independence. When we think about what "independent" means; it means that every vector is independent of all the others. In other words it cannot be written as a linear combination of other vectors.

Answer 3

To build intuition about the notion of linearly independent vectors, look first at what happens when there is a finite number of vectors in the set $S$.

Then, start from an empty new set $T$ and add the vectors of $S$ one by one: the set $S$ is linearly independent iff each time that you add a vector (from $S$) to $T$ then the vector space generated by $T$ is strictly larger than at the previous iteration.

The order in which you add the vectors is not important, because generating the vector space from $T$ means taking all the linear combinations of the vectors in $T$. Thanks to the associativity and distributivity properties of sum and scalar product, you see that you get the same generated spaces whatever the order.

The negation of linearly independent, then, is that at some point when you add a vector (from $S$) to $T$, your resulting space generated from $T$ (by all linear combinations) does not increase at all. And that means that the added vector was already in the vector space generated from $T$ at the previous step.

So that means exactly, in the most intuitive and obvious way, that the vector (the one added at that step) is a linear combination of others in the set.

This example is taking simplified hypotheses, so the reasoning would be different with, say, vector spaces of infinite dimension, but at least you see the mechanics and the motivation behind these notions.